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SUMMARY TECHNICAL REPORT 
OF THE 

NATIONAL DEFENSE RESEARCH COMMITTEE 


This document contains information affecting the national defense of the 
United States within the meaning of the Espionage Act, 50 U.S.C., 31 and 32, 
as amended. Its transmission or the revelation of its contents in any manner 
to an unauthorized person is prohibited by law. 

This volume is classified CONFIDENTIAL in accordance with security regu- 
lations of the War and Navy Departments because certain chapters contain 
material which was CONFIDENTIAL at the date of printing. Other chapters 
may have had a lower classification or none. The reader is advised to consult 
the War and Navy agencies listed on the reverse of this page for the current 
classification of any material. 


Manuscript and illustrations for this volume were prepared for 
publication by the Summary Reports Group of the Columbia 
University Division of War Research under contract OEMsr-1131 
with the Office of Scientific Research and Development. This vol- 
ume was printed and bound by the Columbia University Press. 

Distribution of the Summary Technical Report of NDRC has 
been made by the War and Navy Departments. Inquiries concern- 
ing the availability and distribution of the Summary Technical 
Report volumes and microfilmed and other reference material 
should be addressed to the War Department Library, Room 
lA-522, The Pentagon, Washington 25, D. C., or to the Office of 
Naval Research, Navy Department, Attention: Reports and 
Documents Section, Washington 25, D. C. 


Copy No. 

172 


This volume, like the seventy others of the Summary Technical 
Report of NDRC, has been written, edited, and printed under 
great pressure. Inevitably there are errors which have slipped past 
Division readers and proofreaders. There may be errors of fact not 
known at time of printing. The author has not been able to follow 
through his writing to the final page proof. 

Please report errors to: 

JOINT RESEARCH AND DEVELOPMENT BOARD 
PROGRAMS DIVISION (STR ERRATA) 

WASHINGTON, 25, D. C. 

A master errata sheet will be compiled from these reports and sent 
to recipients of the volume. Your help will make this book more 
useful to other readers and will be of great value in preparing any 
revisions. 



SUMMARY TECHNICAL REPORT OE THE 
APPLIED MATHEMATICS PANEL, NDRC 

VOLUME 3 


PROBABILITY AND STATISTICAL 
STUDIES IN WARFARE 
ANALYSIS 


OFFICE OF SCIENTIFIC RESEARCH AND DEVELOPMENT 
VANNEVAR BUSH, DIRECTOR 

NATIONAL DEFENSE RESEARCH COMMITTEE 
JAMES B. CONANT, CHAIRMAN 

APPLIED MATHEMATICS PANEL 
WARREN WEAVER, CHIEF 


WASHINGTON, D. C., 1946 


NATIONAL DEFENSE RESEARCH COMMITTEE 


James B. Coiiant, Cliairntan 
lliehard C. Tolman, Vice Chairman 
Boger Adams Army Eepreseiitative^ 

Frank B. JeM^ett Navy Eepresentative- 

Karl T. Compton Commissioner of Patents^ 

Irvin SteM^art, Executive Secretary 


'^Army representatives in order of service: 

Maj. Gen. G. V. Strong Col. L. A. Denson 

Maj. Gen. R. C. Moore Col. P. R. Faymonville 

Maj. Gen. C. C. Williams Brig. Gen. E. A. Regnier 

Brig. Gen. W. A. Wood, Jr. Col. M. M. Irvine 

Col. E. A. Routheau 


^Navy representatives in order of service: 

Rear Adm. H, G. Bowen Rear Adm. J. A. Furer 

Capt. Lybrand P. Smith Rear Adm. A. H. Van Keuren 

Commodore H. A. Schade 
^Commissioners of Patents in order of service: 

Conway P. Coe Casper W. Ooms 


NOTES ON THE OEGANIZATION OF NDEC 


The duties of the National Defense Research Committee were 
(1) to recommend to the Director of OSRD suitable projects 
and research programs on the instrumentalities of warfare, 
together with contract facilities for carrying out these projects 
and programs, and (2) to administer the technical and scien- 
tific work of the contracts. More specifically, NDRC func- 
tioned by initiating research projects on requests from the 
Army or the Navy, or on requests from an allied government 
transmitted through the Liaison Office of OSRD, or on its 
own considered initiative as a result of the experience of its 
members. Proposals prepared by the Division, Panel, or 
Committee for research contracts for performance of the 
work involved in such projects were first reviewed by NDRC, 
and if approved, recommended to the Director of OSRD. 
Upon approval of a proposal by the Director, a contract per- 
mitting maximum flexibility of scientific effort was arranged. 
The business aspects of the contract, including such matters 
as materials, clearances, vouchers, patents, priorities, legal 
matters, and administration of patent matters were handled 
by the Executive Secretary of OSRD. 

Originally NDRC administered its work through five di- 
visions, each headed by one of the NDRC members. These 
were : 

Division A — Armor and Ordnance 

Division B — Bombs, Fuels, Gases, & Chemical Problems 
Division C — Communication and Transportation 
Division D — Detection, Controls, and Instruments 
Division E — Patents and Inventions 


In a reorganization in the fall of 1942, twenty-three ad- 
ministrative divisions, panels, or committees were created, 
each with a chief selected on the basis of his outstanding work 
in the particular field. The NDRC members then became a 
reviewing and advisory group to the Director of OSRD. The 
final organization was as follows: 

Division 1 — Ballistic Research 

Division 2 — Effects of Impact and Explosion 

Division 3 — Rocket Ordnance 

Division 4 — Ordnance Accessories 

Division 5 — New Missiles 

Division 6 — Sub-Surface Warfare 

Division 7 — Fire Control 

Division 8 — Explosives 

Division 9 — Chemistry 

Division 10 — Absorbents and Aerosols 

Division 11 — Chemical Engineering 

Division 12 — Transportation 

Division 13 — Electrical Communication 

Division 14 — Radar 

Division 15 — Radio Coordination 

Division 16 — Optics and Camouflage 

Division 17 — Physics 

Division 18 — War Metallurgy 

Division 19 — Miscellaneous 

Applied Mathematics Panel 

Applied Psychology Panel 

Committee on Propagation 

Tropical Deterioration Administrative Committee 



•V 




NDRC FOREWORD 


AS EVENTS of the years preceding 1940 revealed 
more and more clearly the seriousness of the 
world situation, many scientists in this country came 
to realize the need of organizing scientific research 
for service in a national emergency. Recommenda- 
tions which they made to the White House were 
given careful and sympathetic attention, and as a 
result the National Defense Research Committee 
[NDRC] was formed by Executive Order of the Pres- 
ident in the summer of 1940. The members of NDRC, 
appointed by the President, were instructed to sup- 
plement the work of the Army and the Navy in the 
development of the instrumentalities of war. A year 
later, upon the establishment of the Office of Scien- 
tific Research and Development [OSRD], NDRC 
became one of its units. 

The Summary Technical Report of NDRC is a 
conscientious effort on the part of NDRC to summa- 
rize and evaluate its work and to present it in a useful 
and permanent form. It comprises some seventy vol- 
umes broken into groups corresponding to the NDRC 
Divisions, Panels, and Committees. 

The Summary Technical Report of each Division 
Panel, or Committee is an integral survey of the work 
of that group. The first volume of each group’s re- 
port contains a summary of the report, stating the 
problems presented and the philosophy of attacking 
them, and summarizing the results of the research, 
development, and training activities undertaken. 
Some volumes may be “state of the art” treatises 
covering subjects to which various research groups 
have contributed information. Others may contain 
descriptions of devices developed in the laboratories. 
A master index of all these divisional, panel, and com- 
mittee reports which together constitute the Sum- 
mary Technical Report of NDRC is contained in a 
separate volume, which also includes the index of a 
microfilm record of pertinent technical laboratory 
reports and reference material. 

Some of the NDRC-sponsored researches which 
had been declassified by the end of 1945 were of suffi- 
cient popular interest that it was found desirable 
to report them in the form of monographs, such as 
the series on radar by Division 14 and the mono- 
graphs on sampling inspection by the Applied Mathe- 
matics Panel. Since the material treated in them is 


not duplicated in the Summary Technical Report of 
NDRC, the monographs are an important part of 
the story of these aspects of NDRC research. 

In contrast to the information on radar, which is 
of widespread interest and much of which is released 
to the public, the research on subsurface warfare is 
largely classified and is of general interest to a more 
restricted group. As a consequence, the report of 
Division 6 is found almost entirely in its Summary 
Technical Report, which runs to over twenty vol- 
umes. The extent of the work of a division cannot 
therefore be judged solely by the number of volumes 
devoted to it in the Summary Technical Report of 
NDRC: account must be taken of the monographs 
and available reports published elsewhere. 

Perhaps the highest tribute which could have been 
paid to the role of mathematicians in World War II 
was the complete lack of astonishment which greeted 
their contributions. To the Applied Mathematics 
Panel of NDRC came urgent, varied, and formidable 
requests from every other group in NDRC and every 
military service. As expected, these requests were 
met; and, also as expected, the results were found 
invaluable in every phase of warfare from defense 
against enemy attack to the design of new weapons, 
recommendations for their use, predictions of their 
usefulness, and analysis of their effects. 

To meet such obligations, the Applied Mathemat- 
ics Panel under the leadership of Warren Weaver, to- 
gether with members of its staff and of its con- 
tractors’ staffs, made available the services of a group 
of men who were not merely able, competent mathe- 
maticians but also loyal, devoted Americans co- 
operating unselfishly in the defense of their country. 
The Summary Technical Report of the Applied 
Mathematics Panel, prepared under the direction of 
the Panel Chief and authorized by him for publica- 
tion, is a record of their accomplishments and a testi- 
monial to their scientific integrity. They deserve the 
grateful appreciation of the Nation. 

Vannevar Bush, Director 
Office of Scientific Research and Development 

J. B. CoNANT, Chairman 
National Defense Research Committee 


V 




FOREWORD 


W HEN THE National Defense Research Committee 
was reorganized at the end of 1942, it was de- 
cided to set up a new organization, called the Applied 
Mathematics Panel [AMP], in order to bring mathe- 
maticians as a group more effectively into the work 
being carried on by scientists in support of the na- 
tion’s war effort. At the time of the original appoint- 
ment of the National Defense Research Committee 
by President Roosevelt, no mathematicians were in- 
cluded on the Committee, and it was not until the 
NDRC had been operating for more than a year that 
the need of a separate division devoted to applied 
mathematics was recognized. Although many of the 
operating Divisions of NDRC had set up mathe- 
matical groups to handle their own analytical prob- 
lems, it was intended that the new Applied Mathe- 
matics Panel should supplement such groups and 
should furnish mathematical advice and service to 
all Divisions of the NDRC, carrying out requested 
mathematical analyses and remaining available as 
consultants after the original analyses had been com- 
pleted. The Panel was organized too late to make 
possible a fully definitive trial of the success of this 
type of organization. That mathematics has a funda- 
mental role to play in the science of warfare, I am 
sure; I have set forth some of the considerations 
which seem to me relevant and important in the last 
chapter of Volume 2 of the AMP Summary Tech- 
nical Report. 

The actual development of wartime scientific work 
proved to be such that the Applied Mathematics 
Panel has not only been called upon for assistance by 
NDRC Divisions but has also directly assisted many 
branches of the Army and Navy. Indeed, at the 
conclusion of hostilities, when approximately two 
hundred studies had been undertaken by the Panel, 
roughly one-half of these represented direct requests 
from the Armed Services. Furthermore, the consult- 
ing activities, growing out of studies originally under- 
taken to answer specific questions, turned out to be 
considerably more extensive and significant than was 
originally anticipated. I think that the importance of 
this phase of the work cannot be too strongly em- 
phasized. But no account of such general consulting 
activities is given here, this report being restricted to 
the formally constituted studies. 

The analytical work under AMP studies was car- 
ried on by mathematicians associated in groups at 
various universities and operating under OSRD con- 
tracts administered by the Panel. To the men who 


served as technical representatives of the universi- 
ties under these contracts, and to the technical aides 
who assisted the Chief in the administration of the 
Panel’s scientific work, the Panel owes a large measure 
of whatever success it achieved. These men combined 
outstanding scientific competence with energy, re- 
sourcefulness, and a selfless willingness to devote 
their own efforts, as well as the efforts of their staffs, 
to the solution of other people’s problems. The general 
plans for the Panel’s activities were based upon the 
counsel of a group of eminent mathematicians, for- 
mally labeled the Committee Advisory to the Scientific 
Officer. This group, meeting every week and con- 
sisting of R. Courant, G. C. Evans, T. C. Fry, 
L. M. Graves, H. M. Morse, 0. Veblen, and S. S. 
Wilks, had responsibility for the preliminary exami- 
nation of requests which reached the Panel and for 
decisions on overall policy. The Chief relied heavily 
on their advice which, to a large extent, determined 
the effectiveness of the Panel’s activities. 

As the work of NDRC developed, the Panel was 
called upon for assistance by all of NDRC’s nineteen 
Divisions. It is not, therefore, surprising that the 
scope of the Panel’s activities covers a wide range, 
falling into four broad, though somewhat overlap- 
ping, categories: 

1. Mathematical studies based upon certain classical 
fields of applied mathematics, such as classical me- 
chanics and the dynamics of rigid bodies, the theory 
of elasticity and plasticity, fluid dynamics, electro- 
dynamics, and thermodynamics. 

2. Analytical studies in aerial warfare, including 
assessment of the performance of sights and anti- 
aircraft fire control equipment ; studies relating to the 
vulnerability of aircraft to plane-to-plane and to 
antiaircraft fire and the optimal defense of the air- 
plane against these; and analyses of problems arising 
from the use of rockets in air warfare. 

3. Probability and statistical studies concerned with 
the effectiveness of bombing; various aspects of naval 
warfare, including fire effect analysis and the per- 
formance of torpedoes; the design of experiments; 
sampling inspection; and analyses of many types of 
data collected by the Armed Services. 

4. Computational services concerned with the evalu- 
ation of integrals; the construction of tables and 
charts ; the development of techniques adapted to the 
solution of special problems; the nature and capa- 
bilities of computing equipment. 

The work of the Panel in the first two of these 


/: 


(‘DXFTnENTIAI. 




VI ] 


FOREWORD 


viii 


categories is summarized in Volumes 1 and 2 of the 
AMP Summary Technical Report. Volume 3, to- 
gether with two monographs^ which the Panel has 
prepared dealing with sampling inspection and tech- 
niques of statistical analysis, provides a summary of 
the work in the third category. The fourth class of 
activities has been reported in AMP Note 25, 
Description of Mathematical Tables Computed under 
the auspices of the Applied Mathematics Panel, 
NDRC; in AMP Note 26, Report on Numerical 
Methods Employed by the Mathematical Tables Project; 
and in the reports published by the Panel under AMP 
Study 171, Survey of Computing Machines. No at- 
tempt has been made to report on work which will 
shortly be published as articles in scientific journals 
or on results which are deemed too special to be of 
continuing interest. 

^Sampling Inspection and Techniques of Statistical Analysis, 
published by McGraw-Hill Book Co., Inc. 


The preparation of this Summary Technical Re- 
port was undertaken after the end of World War II, 
at a time when the members of the Panel’s staff and 
of the contract groups were eager to return to their 
peacetime careers. Thus the preparation of these 
three volumes, solely for the purpose of recording 
for the Services, in easily accessible form the scien- 
tific results of the Panel’s activities, was achieved 
at real personal sacrifice. I am greatly indebted to 
the authors of the several parts of these volumes and 
to the Editorial Committee, consisting of Mina Rees, 
I. S. Sokolnikoff, and S. S. Wilks, for the admirable 
job they have done in bringing together, under high 
pressure, a summary of the principal scientific ac- 
complishments of the Panel. 

Warren Weaver 
Chief, Applied Mathematics Panel 



PREFACE 


T his volume furnishes a summary of the princi- 
pal results of only a portion of the probability 
and statistical investigations made by the Applied 
Mathematics Panel during World War II. The work 
of the Panel in mathematical statistics may be classi- 
fied into four major categories: (1) bombing accuracy 
studies, (2) development of statistical methods in 
inspection, research, and development work, (3) de- 
velopment of new fire effect tables and diagrams for 
the Navy, and (4) miscellaneous probability and sta- 
tistical studies. As explained by the Chief of the 
Panel in his Foreword, the work done in category (2) 
has been declassified and is being prepared for pub- 
lication in the form of two monograghs; the work in 
category (3) has been transferred to a contract be- 
tween Princeton University and the Navy for con- 
tinuation, and many of the studies under category (4) 
are such that future interest in them is extremely 
limited. 

Accordingly, the probability and statistical work 
of the Panel which is considered appropriate to sum- 
marize in Volume 3 of the Panel’s Summary Tech- 
nical Report consists of that in category (1) together 
with several studies in (4). Volume 3 is therefore di- 
vided into two parts. Part I: Bombing Studies, and 
Part II: Miscellaneous Studies. 

The discussion and material presented in Part I — 
Chapters 1 through 5 — is a resume of the work of the 
Panel on the probability and statistical aspects of 
those bombing studies in which the Panel participated 
on a sufficient scale to warrant distribution of its own 
reports, memoranda, notes, and working papers. Be- 
cause of the diverse interests and requirements of the 
organizations which initiated them, these studies re- 
late to bombing operations of practically every kind, 
including such unusual items as air-to-air bombing, 
clearance of minefields, pitting of airfields, toxic-gas 
bombing, and controlled-missile bombing. In general, 
no attempt is made to cover activity carried out by 
Panel representatives acting as consultants for vari- 
ous agencies, some of which has been fully reported 
by these agencies. There are other omissions, notably, 
discussions of test programs, of operational and prac- 
tice data, of odds-and-ends of theoretical investiga- 
tions carried out by Panel personnel in conjunction 
with others. Many bombing accuracy investigations 
have been carried out by operational analysis sec- 
tions and other groups in the Army and Navy, as well 
as British groups, but no attempt has been made in 


this volume to integrate the work of the Panel into 
the entire field. 

The work has been done at various stages of 
weapon and tactics development, ranging from that 
of pure a priori prognostication to that based fully 
on a posteriori assessment of combat operations. 
Work has rarely been done at the stage of the oper- 
ations analyst, nor has it been practicable to do it, at 
that stage. For, first, the Panel has not been in a 
position to obtain data as quickly as the operations 
analyst — indeed, the Panel has depended on the op- 
erations analyst as one of its prime sources of in- 
formation; secondly, the Panel usually enjoyed more 
liberal time limits and better working facilities than 
did the operations analyst and hence could try for 
solutions to certain problems which were prohibi- 
tively formidable from the viewpoint of the analyst. 
Thus, there was virtually no duplication of effort be- 
tween the Panel on the one hand and the various 
Operations Analysis Sections of the Army Air Forces 
and the Operational Research Group of the Navy on 
the other. It should not be inferred from these com- 
ments on time limits that the Panel work on bombing 
problems was a leisurely pursuit; the time limits, 
while generous compared to those faced by oper- 
ations analysts, were very short when measured 
against the problems posed; indeed, deadlines fre- 
quently compelled that stop-gap solutions be sought, 
and, usually, the pressure of new work precluded an 
aesthetically satisfying clean-up of these problems. 

The methodology of research varied from formal 
mathematical analysis, at one extreme, to synthetic 
processes and statistical experiments or models at the 
other. Formal analysis is the more precise and hence 
satisfying process, but the difficulties of formulating 
the problem in analytical terms and then (worse) of 
finding numerical solutions increase rapidly with the 
complexity of the bombing situation. For example, it 
is very easy to deduce almost all the probability con- 
sequences regarding the problem of aiming a single 
bomb at a rectangular target, but very few deduc- 
tions can be made directly from the equations which 
describe the dropping of a train of as few as three 
bombs on a rectangular target. Since the problem of 
dropping a train of three bombs is itself extremely 
simple, compared to many common bombing oper- 
ations, it is apparent that formal mathematical proc- 
esses cannot alone be depended upon to carry the 
burden, but they are powerful when used in conjunc- 


ix 


X 


PREFACE 


tion with synthetic methods and statistical models. 
These combined methods were being used more and 
more extensively and effectively toward the close 
of the war. 

There has undoubtedly been some waste in the 
Panehs bombing research program, at least judged 
from the short-term viewpoint and with reference to 
the intended applications, for it occasionally hap- 
pened that a great deal of effort was directed toward 
problems which had no large-scale counterpart in 
combat, i.e., toward problems which did not possess 
great potential yield compared to other unsolved 
problems. It is believed that this did not occur fre- 
quently, but it is annoying that it occurred at all. 
Part of it was unavoidable and may be ascribed to 
the natural waste of warfare. The real waste was in- 
curred by continuing large-scale work after combat 
had clearly shown that quite different problems were 
of primary importance. This waste is attributable 
partly to the natural momentum of work under way 
and partly to insufficient liaison with the war the- 
aters ; the latter refers not only to the Paneks liaison, 
but to that of the agencies which requested studies of 
problems. This kind of difficulty was most pronounced 
in the early days of World War II; the situation im- 
proved with time as the Services’ understanding of 
their needs increased and as the Panel’s experience 
broadened to the point where it could better discrim- 
inate between merely unsolved problems and prob- 
lems which were highly pertinent to current, pro- 
jected, or likely operations. 

The work of the Panel in bombing accuracy re- 
search was done by three research groups, namely, 
the Columbia University Bombing Research Group, 
the Princeton University Statistical Research Group, 
and the Statistical Laboratory of the University of 
California. The bombing research work of the Co- 
lumbia group, under the direction of J. Schilt, con- 
sisted primarily in the computation of tables for the 
studies in train bombing and scatter bombing. The 
work of the Princeton group in bombing research, 
directed by J. D. Williams, consisted of a wide va- 
riety of investigations in pattern bombing, toxic gas 
bombing, air-to-air bombing, and so on. The Cali- 
fornia group, under the direction of J. Neyman, 
worked mainly on problems in area bombing, incen- 
diary bombing, and to some extent train bombing. 
Members of these three Panel groups, and in par- 
ticular the Princeton group, worked very closely with 
Army and Navy research groups interested in bomb- 
ing problems. In fact, for periods varying from a few 


months to nearly two years, members of these Panel 
groups acted as consultants on bombing accuracy 
problems to the following agencies: Army Air Forces 
Board; Proving Ground Command, Eglin Field, 
AAF; Combat Analysis Branch, Statistical Control 
Division, AAF; Navy Air Intelligence Group; Joint 
Army-Navy Target Group; Navy Operational Re- 
search Group; and Operational Analysis Division, 
Twentieth Air Force. 

In Part II, Chapters 6, 7, and 8, a summary is pre- 
sented of the principal results of the probability and 
statistical aspects of three torpedo studies, three land 
mine clearance investigations and an extensive sta- 
tistical study of the performance of heat-homing 
devices. This is only a part of an extensive group of 
miscellaneous probability and statistical studies. The 
other studies in this category have been declassified 
or they are so highly specialized as to hold very little 
future interest, and hence are not included in Part 
II. The reader who might possibly be interested in 
such investigations can find brief accounts of the 
facts in the Panel’s Final Summary Report of 
Projects. 

The torpedo studies were done by the Columbia 
University Statistical Research Group and the re- 
maining studies summarized in Part II were made by 
the Princeton University Statistical Research Group. 

Except for a trivially small number of studies 
initiated by the Panel itself, all of the work summa- 
rized in this volume was requested by Army, Navy, or 
NDRC agencies. It would be difficult to give a list 
of all of the agencies with which the Panel had some 
contact with its probability and statistical research 
work described here, but the following, stated with- 
out reference to order, are the ones with which the 
Panel has had the greatest amount of association: 
Divisions 2, 3, 5, 7, 8, and 11 of NDRC; the Army 
Air Forces Board; the Joint Target Group; the Com- 
bat Analysis Branch, Statistical Control Division, 
AAF; the Proving Ground Command, AAF; the 
Armament Laboratory, Wright Field, AAF; the Ball- 
istics Research Laboratory, Aberdeen Proving 
Ground; the Army Engineer Board; the Joint Army- 
Navy Experimental Testing Board; the Navy Air 
Intelligence Group; the Office of the Secretary of 
War; the Operations Analysis Divisions, Twentieth 
Air Force and USASTAF; the Navy Operational 
Research Group; and the Guided Missiles Com- 
mittee of the Joint Chiefs of Staff. 

The Panel is indebted to so many individuals in 
these agencies for information, counsel, and courte- 


/t-'i)N~ri'nKN"iT-Ar^ 


PREFACE 


XI 


sies that it is highly impractical to attempt to list 
their names here. 

In conclusion, the experience of the Panel in bomb- 
ing accuracy analysis and closely related work as 
summarized in this volume indicates that this type 
of analysis is extremely effective in the development 
of weapons and tactics for their employment. It pro- 
vides a powerful scientific method of evaluating the 
effectiveness of a weapon and improving it. Further- 
more, it has become equally apparent from the Panel’s 
experience that this analysis should be carried out 
in an orderly and integrated fashion all the way from 
the original conception of a new type of weapon to 
the use of this weapon in combat. Of course, in peace- 
time it is possible to follow the development of the 
weapon only through the field or proving ground 
testing stage. In this chain of development from con- 
ception to combat, there should be close coordina- 
tion of the mathematical and statistical work on 
weapon accuracy and effectiveness at all stages, i.e., 
original design, development, early testing, advanced 
testing, production, and combat, as well as close co- 
ordination of the agencies involved — scientific, engi- 
neering, and military. 

In view of the implications of the advent of the 
atomic bomb, a large amount of the type of accuracy 
analysis carried out for ordinary bombs becomes ob- 
solete. This factor, however, serves essentially to 
change the emphasis of the work that needs to be 


done from accuracy analysis of ordinary bombs, 
rockets, or gunfire with relatively small radii of 
effectiveness to that of controlled missiles of various 
kinds with extremely large radii of destruction, and 
the accuracy of defensive weapons against such mis- 
siles. It is believed that both the Army and Navy 
will do well to see to it that a carefully coordinated 
program of research of this type is set up and carried 
along in conjunction with the development of new 
weapons, whether they be slight variants of existing 
high explosive bombs or fantastic new controlled 
missiles with atomic payloads. 

Finally, the editor of this volume wishes to express 
the thanks of the Panel and his own gratitude to 
J. D. Williams, Technical Aide of the Panel, for pre- 
paring the major portion of the volume, namely, 
Part I. He has done an excellent job under high 
pressure and a difficult deadline in bringing together 
a summary, necessarily rather highly condensed, of 
the principal accomplishments of the Panel in bomb- 
ing research. He is uniquely qualified to do this work 
since he has played a central role in the Panel’s 
bombing research work. Part II was prepared by the 
editor of the volume in consultation with various 
members of the Columbia University Statistical Re- 
search Group and the Princeton University Statis- 
tical Research Group. 

S. S. Wilks 
Editor 





CONTENTS 


CHAPTEK PAGE 

Summary by ^Yan'en ^Y eaver 1 

PART I 

BOMBING STUDIES 

1 General Considerations 9 

2 Single-Eelease Bombing 14 

3 Train Bombing 23 

4 Pattern Bombing 46 

5 Further Investigations 58 

PART II 

MISCELLANEOUS STUDIES 

6 Torpedo Studies 69 

7 Statistical Studies in Mine Clearance 79 

8 Statistical Analysis of the Performance of Heat-Homing 

Devices 88 

Bibliography 93 

OSPD Appointees 96 

Contract Numbers 97 

Service Project Numbers 99 


Index 


101 




SUMMARY 


I N THIS Suminaiy Technical Report of the Applied 
Mathematics Panel, a resume is given of the prin- 
cipal scientific accomplishments of the Panel from its 
beginning in 1943 until the conclusion of hostilities. 
The activities here reported cover a wide range, deal- 
ing as they do with studies undertaken at the request 
of each of the nineteen Divisions of NDRC and of 
many branches of the Army and Navy. For the pur- 
pose of this report, that portion of the Panel’s work 
which deals with specific military problems has been 
divided into three parts: Volume 1, Mathematical 
Studies Relating to Military Physical Research; Vol- 
ume 2, Analytical Studies in Aerial Warfare; and 
Volume 3, Probability and Statistical Studies in War- 
fare Analysis. In addition to reporting on specific 
military problems. Volume 1 also indicates direc- 
tions in which certain of the theories of fluid dynam- 
ics have been extended under AMP auspices as an 
aid in the planning and interpretation of military 
experiments, and in understanding the operation of 
enemy weapons. These three volumes contain no 
account of the new developments in statistical meth- 
ods which have already been partially reported in a 
published article^ and a published book^ on sequen- 
tial analysis, nor of certain important new applica- 
tions of statistical theory which grew out of the 
Panel’s attempt to solve problems presented to it by 
the Services. These latter are reported in two pub- 
lished monographs. Sampling Inspection and Tech- 
niques of Statistical Analysis^ prepared under Panel 
auspices, which form part of the Panel’s report of 
its technical activities. (Published by McGraw-Hill.) 

Most AMP studies were concerned with the im- 
provement of the theoretical accuracy of equipment 
by suitable changes in design; or with the develop- 
ment of basic theory, particularly in the field of fluid 
dynamics; or with the best use of existing equipment, 
particularly in fields like bombing and the barrage 
use of rockets. Two studies carried out under AMP 
auspices come closer to having general tactical or 
strategic scope than do most of the other work. I 
have myself given an account of these two studies 
in Part IV of Volume 2, where I have also set forth 
some incomplete and preliminary ideas of what a 
general analytical theory of air warfare could and 
should comprise and some arguments for and against 


®By Warren Weaver 


attempting to construct and use such a theory. I have 
there indicated how certain activities of the Applied 
Mathematics Panel and of other agencies relate to a 
scheme for a broad approach to the problems of air 
warfare and of warfare in general, and I have pointed 
out some of the contributions which mathematics can 
make to the field of national defense. 

That part of the Panel’s work which may be roughly 
described as classical applied mathematics is pre- 
sented in Volume 1. Certain phases of this subject were 
developed under Panel auspices and adapted to prob- 
lems of military interest, the principal emphasis be- 
ing on problems of primary concern to the Navy. 

In the early stages of the war, certain acoustic 
equipment employed in submarine detection by echo 
ranging used a “dome” — a streamlined convex shell 
filled with water or other liquid, such as oil. The 
presence of these domes caused interference with the 
directional pattern sent out from the projector, and 
in some of the equipment the disturbance was ex- 
tremely serious. The Panel was asked to study the 
situation and to suggest changes in the domes which 
would minimize the disturbances. Practical con- 
clusions were reached regarding desirable materials 
and design. It was found desirable for practical 
reasons to use thin shells reinforced by stiffening 
elements such as ribs and rods rather than to achieve 
strength by general thickness. Difficulties arising in 
direction finding due to annoying reflections were 
also analyzed, and suggestions were made for im- 
proving conditions, for example, by corrugations on 
the inner surface of the side walls of the domes. This 
dome study was one aspect of the work in wave 
propagation with which the Panel was concerned. 
There were others. For example, an investigation 
was made of the scattering of electromagnetic waves 
by spherical objects to assist in the analysis of smokes 
and fogs. A study of somewhat similar mathematical 
character (but dealing with electromagnetic disturb- 
ances rather than actual mechanical waves in a 
liquid) was undertaken at the request of the Fire 
Control Division (Division 7, NDRC), which had 
under development a predictor, the T-28, intended 
for use with the 40-mm gun. The computing mechan- 
ism used by this predictor included a sphere on which 
were placed electrical windings in such a way that 
the resulting field was one which corresponded to one 
simple dipole at the center of the sphere. Although 




1 


2 


SUMMARY 


the theoretical way in which the winding should be 
distributed on the surface of this sphere was well 
known, it was necessary as a practical matter to 
substitute a winding in which the turns were located 
in grooves on the sphere. The formulas resulting from 
the Panel’s study of this problem form a basis for 
practical applications which include ammeters, 
galvanometers, and direction finders. This mathe- 
matical study was of critical importance for the fire 
control instrument in question, for without it, it was 
impossible to obtain useful accuracy in the spherical 
‘‘electromagnetic resolver” which carried out the 
essential steps in the target predicting process. 

The Panel’s work in gas dynamics, mechanics, and 
underwater ballistics is also reported in this first 
volume. The Panel’s work in gas dynamics was prin- 
cipally concerned with the theory of explosions in the 
air and under water, and with certain aspects of jet 
and rocket theory. New developments were made in 
the study of shock fronts, associated with violent 
disturbances of the sort which result from explosions. 
An interesting and significant aspect of the work was 
concerned with Mach phenomena which frequently 
play a practical role in determining the destructive 
effects of shocks. For example, the advantages of air- 
bursting large blast bombs were suggested by a con- 
sideration of Mach waves. A request from the Bureau 
of Aeronautics for assistance in the design of nozzles 
for jet motors to be used for assisted take-off gave 
rise to an extended study of gas flow in nozzles and 
supersonic gas jets. As a result, suggestions were made 
not only for the design of nozzles for jet-assisted take- 
off, but also for “perfect” exhaust nozzles and com- 
pressors (of use in supersonic wind tunnels) and for 
various instruments to aid in rocket development and 
experimentation. The jet propulsion studies were re- 
lated to Army and Navy interest in intermittent jet 
motors of the V-1 type. Jet propulsion under water 
was also studied, with results which should prove use- 
ful as a guide to experiment in this field where experi- 
mentation has thus far not reached the stage where 
the theoretical results can be fully put to test. 

The problems in mechanics fall under two general 
headings: (1) those involving the mechanics of par- 
ticles and rigid bodies and (2) those involving the 
mechanics of a continuum. For example, a study in 
the second category sought possible explanations of 
the break-up in cylindrical powder grains in the 
43/^-in. rocket to explain difficulties which were being 
encountered at the Allegheny Ballistics Laboratory, 
and an experimental program was outlined for the 


testing of the most probable theories. One of the most 
interesting of the mechanical studies concerned the 
so-called spring hammer box used by the U. S. Navy 
in acoustic mine warfare. The dependence of the 
operation of this device on various physical param- 
eters (for example, the mass of the hammer) was 
analyzed with the aid of a simple mechanical model, 
and of an electrical analog. Another problem of this 
type studied the dynamics of the gun equilibrator, or 
balancing system, when an Army gun was mounted 
on board a ship. The pitching and rolling of the ship 
naturally introduced special difficulties. 

In the section on underwater ballistics, the prob- 
lems involved are classified according to the various 
phases in the motion of the projectile: the impact 
phase, the development of the cavity, and the under- 
water trajectory. During the impact phase, forces act 
which are important partly because of their possible 
effects on the nose structure and mechanism of the 
projectile, partly because of their influence in deter- 
mining the projectile’s subsequent motion. It is dur- 
ing the impact phase that the greatest deceleration 
occurs. The theoretical analysis involves, among 
many other considerations, the direction of entry 
(vertical or oblique), and the shape of the projectile. 
Save when the speed of a missile is slow, its entry is 
accompanied by the formation of a cavity which be- 
comes sealed behind the projectile and accompanies 
it to a greater or less extent during its underwater 
motion, influencing that motion in an important way. 
The underwater trajectory itself presents problems 
of great complexity. Frequently, slight changes in 
values of the parameters which determine the motion 
will cause a complete change in the type of motion. 
A mathematical discrimination among the several 
types of motion is made, part of the distinction de- 
pending on such things as the position of the center 
of gravity of the missile, the ratio of its length to its 
diameter, its density, its radius of gyration, and the 
manner of its entry. Throughout this treatment, an 
attempt has been made to integrate into a single re- 
port the results which have been obtained by the 
many agencies concerned with the several phases of 
the problem and thus to assist the theoretical and 
experimental studies which must be carried forward 
in future attempts to understand this difficult array 
of problems. 

Many of the studies reported in Volume 2, as well 
as those contained in Volume 3, involve probability 
considerations, a field which is notoriously tricky and 
within which “common sense” is often quite helpless. 


jTnXFJTtirNTD 


SUMMARY 


3 


For example, what is the optimum mixture of armor- 
piercing and incendiary ammunition for the roar guns 
of a bomber? Specifications often designate such 
mixtures as five AP to two incendiary (we are 
neglecting tracers here). Why? The somewhat strik- 
ing, and by no means obvious, fact is that, given any 
fixed type of target, it is better to have either all AP 
or all incendiary, depending on the nature of the 
target. The justification for any other intermediate 
mixture should be based on knowledge of the relative 
probability of encountering different targets, certain 
of which would be more vulnerable to AP and others 
more vulnerable to incendiary. This conclusion was 
reached as an incidental result of a study which was 
concerned with alternative fighter-plane armament 
and which arose out of the enthusiasm of a few per- 
sons associated with the Panel for two papers at- 
tributable to L. B. C. Cunningham, Chief of the Air 
Warfare Analysis Section in England, and his as- 
sociates. Another study concerned with the practical 
effectiveness of equipment grew out of a request to 
NDRC from Headquarters, AAF, asking for collabo- 
ration with the AAF “in determining the most ef- 
fective tactical application of the B-29 airplane.” 
The results of this study, obtained on the basis of 
large-scale experiments in New Mexico and small- 
scale optical experiments by the Mt. Wilson Ob- 
servatory staff at Pasadena, were concerned princi- 
pally with the defensive strength of single B-29’s and 
of squadrons of B-29’s against fighter attack, and the 
effectiveness of fighters against B-29’s. One indirect 
result of the optical studies was a set of moving 
pictures showing the fire power variation of forma- 
tions as a fighter circles about them. Concerning such 
pictures the President of the Army Air Forces Board 
remarked that he “believed these motion pictures 
gave the best idea to airmen as to the relative effect 
of fire power about a formation yet presented.” Cer- 
tain of these pictures were flown to the Marianas and 
viewed by General Lc May and by many gunnery 
officers at the front. 

These two studies are reported in the last part of 
Volume 2. The first three parts of this volume report 
on special and detailed problems which arise when 
shots are fired against targets moving in the air or on 
the ground. The problem of shooting from an aircraft 
in motion against an enemy aircraft or against a 
ground target in motion and the problem of shooting 
from the ground or from a naval craft against an 
enemy aircraft all involve a number of considerations. 

1. Whenever the target is in motion, its position at 


the instant of firing is different from its position at 
impact, if impact occurs. For an effective shot, the 
motion of the target during the time of flight of the 
bullet or rocket or shell must therefore be predicted, 
at least approximately. The special character of this 
problem for the special cases which have come under 
the PaneUs study are discussed for air-to-air warfare 
in Part I, for rocket fire from the air in Part II, and for 
ground or ship based antiaircraft fire in Part III. 

2. When one’s own ship is in motion, the apparent 
motion of the target is affected. 

3. There are oscillations in aim as the gunner at- 
tempts to point continuously at the target. These 
oscillations are greater in air-to-air and in ship-to-air 
than in ground-to-air gunnery because of the vibra- 
tions, rotations, and bumpy motions of one’s own ship. 

4. There is the effect of gravity on the bullet. In 
air-to-air gunnery, for the short ranges used in World 
War II, this was on minor importance, but for rocket 
fire it introduced very considerable complications. 

5. The resistance of the air varies with the altitude. 
Thus, at 22,000 feet above sea level the air is half as 
dense as it is at sea level. This will affect the average 
speed of a bullet, hence its time of flight, and hence 
the prediction referred to above. 

A large part of Volume 2 is devoted to problems 
connected with so-called flexible gunnery, i.e., with 
the aiming of those guns, carried on aircraft, which 
can be pointed in various directions with respect to 
the aircraft (as contrasted with fixed guns in the 
wings or nose, which are aimed only by movement 
of the aircraft). In January 1944, Brigadier General 
Robert W. Harper, AC /AS (Training), wrote in a 
letter to Dr. Vannevar Bush, Director of OSRD, that 
“the problems connected with flexible gunnery are 
probably the most critical being faced by the Air 
Forces today. It would be difficult to overstate the 
importance of this work or the urgency of the need; 
the defense of our bomber formations against fighter 
interception is a matter which demands increasing 
coordinated expert attention.” This situation arose 
because of the inadequate training and inadequate 
deflection rules given to the gunners who had to 
handle ring sights in bombers. The “relative speed” 
and “apparent motion” rules currently taught were 
not thoroughly learned by the gunners and in many 
cases were by no means adequate when they were 
properly applied. There were well authenticated 
cases of gunners who “led” the attacking fighters in 
a direction exactly opposite to that of the true lead! 

The immediate proposal contained in General 


4 


SUMMARY 


Harper’s letter was that the Applied Mathematics 
Panel should recruit and train competent mathe- 
maticians who had the “versatility, practicality, and 
personal adaptability requisite for successful service 
in the field;” it was planned that these men, after 
two months’ training in this country, would be as- 
signed to the Operations Research Sections in the 
various theaters to devote their attention to aerial 
flexible gunnery problems. The Panel was in a posi- 
tion to carry out this program because it had already 
been drawn into studies of rules for flexible gunnery 
training and because it had access to many of the 
ablest young mathematicians in the country. The 
assignment was completed promptly, and, as a 
partial result of this undertaking, the Panel found 
itself even more closely in touch with the Operations 
Analysis Division of the AAF (with which it had 
already established cordial working relations) and 
with the AAF Central School for Flexible Gunnery. 
Around this interest and the interest of the Army, 
the Navy, Division 7, and Division 14 in the im- 
provement in the effectiveness of guns as well as 
gunnery, grew up a very considerable body of knowl- 
edge and experience which is reported in Part I of 
Volume 2. Here an attempt is made to bring together 
into a single account the state of the art of air-to-air 
gunnery, not only as that has been affected by the 
work of the Applied Mathematics Panel, but as it 
has reflected the activities of agencies in this country 
and abroad. The topics discussed are: 

1. The motion of a projectile from an airborne gun, 
constituting that branch of exterior ballistics which 
is called aeroballistics. 

2. A mathematical theory of deflection shooting con- 
sidered first for the case of a target moving at con- 
stant speed on a straight line which lies in a plane 
with the gun-mount velocity vector; second, for a 
target which moves in a curved path; and third, for 
the case where mount and target move in arbitrary 
space paths. 

3. Pursuit curve theory. Pursuit curves were im- 
portant in World War II, since the standard fighter 
employed a heavy battery of guns so fixed in the air- 
craft as to fire sensibly in the direction of flight. Thus 
it was necessary to fly on such a correctly banked 
turn that a correct and changing aiming allowance 
was continuously made. This pursuit curve theory is 
also of importance in the study of guided missiles 
which continuously change direction under radio, 
acoustic, or optical guidance unwillingly supplied by 
the target. 


4. The design and characteristics of own-speed 
sights which were introduced as devices designed for 
use against the special case of pursuit curve attack 
on a defending bomber. Simple charts which might 
be used in the air are given, based on optimum rules 
for determining deflection against an aerodynamic 
pursuit curve. 

5. Lead computing sights which do not assume that 
the fighter is coming in on a pursuit curve but which 
basically assume that the target’s track relative to 
the gun mount is essentially straight over the time 
of flight of the bullet. The mechanical sights of the 
Sperry series are considered in some detail. 

6. The basic theory of a central station fire control 
system. 

7. The analytical aspects of experimental programs 
for testing airborne fire control equipment. It is recog- 
nized that field tests, laboratory tests, and theoretical 
analyses all have an important place in such a pro- 
gram. Instrumentation for tests, reduction of data, 
measures of effectiveness, and optimum dispersion 
are discussed. 

8. New developments, such as stabilization and the 
use of radar. 

The second part of Volume 2 is devoted largely to a 
presentation of the results obtained by the Panel in a 
study intended to determine what sighting methods 
are feasible for airborne rockets. The essential prob- 
lems involved in this question have to do with ballis- 
tic formulas, attack angle and skid, the effect of wind 
and target motion, how these various factors affect 
each proposed sighting method, and how tracking 
affects and is affected by them. 

In Part III of Volume 2 certain special studies of 
antiaircraft equipment which were made under AMP 
auspices are discussed, and a report is given of the 
flak analysis and other fragmentation and damage 
studies carried on by the Panel. This report is con- 
cerned with some mathematical problems which 
arise in attempts to estimate the probability of 
damage to an aircraft or group of aircraft from one 
or many shots from heavy antiaircraft guns. Related 
problems arise in air-to-air bombing and in air-to-air 
or ground-to-air rocket fire, but the major part of the 
mathematical analysis so far performed has been 
devoted to problems of flak risk. The emphasis in the 
discussion is on the description of a method for treat- 
ing problems of risk, since specific numerical con- 
clusions are likely to become obsolete before further 
need for them arises, while the techniques by which 
the results were obtained will be useful as long as 


SUMMARY 


5 


weapons which destroy by means of flying fragments 
are in use. The original experimental information on 
which the Panel computations were based came from 
a variety of sources, principally Army, Navy, OSRD, 
and British reports. The Panehs chief contribution 
was the development of computational techniques 
which could be carried through before the project be- 
came obsolete, the selection of pertinent examples, 
and the applications of the computational tech- 
niques to the selected examples. Certain applications 
of the underlying theory to time-fuzed and proximity- 
fuzed shells, and to proximity-fuzed rockets are here 
reported. 

Another major field of effort in the work of the 
Panel is that of Mathematical Statistics, reported in 
Volume 3. A remarkably wide variety of probability 
and statistical investigations was carried out by the 
Panel. These investigations ranged from the devel- 
opment of sampling inspection plans in connection 
with procurement of military materiel to extensive 
statistical analyses of combat data. Of the Panel’s 
194 studies, 53 related to problems in probability 
and statistical analysis. 

The work of the Panel in mathematical statistics 
can be grouped into the following major categories: 

1. Bombing accuracy research. 

2. Development of statistical methods in inspection, 
research, and development work. 

3. Development of new fire effect tables and dia- 
grams for the Navy. 

4. Miscellaneous studies relating to spread angles 
for torpedo salvos, lead angles for aerial torpedo 
attacks against maneuvering ships, land mine clear- 
ance, performance of heat-homing devices, search 
problems, verification of weather forecasting for mili- 
tary purposes, procedures for testing sensitivity of ex- 
plosives, distribution of Japanese balloon landings, etc. 

Of these four main categories of work, category 1 
required by far the greatest amount of energy. This 
activity had its beginning in a fairly small study 
undertaken for the Armament Laboratory, AVright 
Field, on the design of a computer for determining the 
optimum spacing of bombs in a train of bombs 
dropped from a bomber in attacking a given target 
under specified conditions. The study was started in 
1942 under Division 7, NDRC, and was transferred 
to the Panel when the Panel was organized. In pur- 
suing this study the group working on it came in 
contact with individuals in more than a dozen Army, 
Navy, and NDRC groups interested in bombing ac- 
curacy problems. As the war progressed, an increas- 


ing number of requests came from these groups for 
studies of all kinds of accuracy and coverage prob- 
lems arising in train bombing, area bombing, pattern 
bombing, guided-missile bombing, incendiary bomb- 
ing, and so on. By the end of the war the work in this 
field had grown to the point where the major effort 
of three Panel research groups was being spent on 
nineteen studies dealing with probability and statisti- 
cal aspects of bombing problems. 

The methods and results developed in category 2 
are of much broader interest than that associated 
with their wartime applications. During the war, it 
was recognized by the Services that the statistical 
techniques which were developed by the Panel for 
Army and Navy use, on the basis of the new theory 
of sequential analysis, if made generally available to 
industry, would improve the quality of products pro- 
duced for the Services. In March 1945, the Quarter- 
master General wrote to the War Department liaison 
officer for NDRC a letter containing the following 
statement : 

By making this information available to Quartermaster 
contractors on an unclassified basis, the material can be 
widely used by these contractors in their own process control 
and the mere process quality control contractors use, the 
higher quality the Quartermaster Corps can be assured of ob- 
taining from its contractors. For, by and large, the basic cause 
of poor quality is the inability of the manufacturer to realize 
when his process is falling down until he has made a consider- 
able quantity of defective items With thousands of con- 

tractors producing approximately billions of dollars’ worth of 
equipment each year, even a 1% reduction in defective mer- 
chandise would result in a great saving to the Government. 
Based on our experience with sequential sampling in the past 
year, it is the considered opinion of this office that savings of 
this magnitude can be made through wide dissemination of 
sequential sampling procedures. 

On the basis of this and similar requests, the 
Panel’s work on sequential analysis was declassified, 
and the reports mentioned above were published. The 
Quartermaster Corps reported in October 1945 that 
at least 6,000 separate installations of sequential 
sampling plans had been made and that in the few 
months prior to the end of the war new installations 
were being made at the rate of 500 per month. The 
maximum number of plans in operation simultane- 
ously was nearly 4,000. 

Thus extensive use was made by the Army of 
sequential analysis as a basis for sampling inspection. 
It was at the request of several Navy bureaus that 
the Panel undertook to assemble a manual setting 
forth procedures to be used not only with sequential 
sampling but also with single and double sampling 


6 


SUMMARY 


plans. As an extension and expansion of this manual, 
the Panel undertook the preparation of its mono- 
graph, Sampling Inspection. The monograph. Tech- 
niques of Statistical Analysis, presents a variety of 
statistical methods which have been developed, or 
adapted from more general methods, for dealing with 
various statistical problems which have arisen in 
connection with research and development work. 

The work done in category 3 was of highly special- 
ized long-range interest to the Office of the Com- 
mander in Chief of the U. S. Fleet. After the work 
had been carried forward under the direction of the 
Panel for nearly two years, arrangements were made 
to transfer and continue the work under a contract, 
effective June 1, 1945, between the Navy and Prince- 
ton University. During the time this work was under 
the Panel’s direction, a series of nine basic reports 
was submitted to the Navy. None of this work, which 
was only partially completed under the direction of 
the Panel, is reported upon in the Panel’s Summary 
Technical Report. 

Certain of the studies in category 4 are of such 
limited interest that it has been considered neither 
appropriate nor worth-while to report upon them 
here. Accounts are given of the work which relates 
to torpedoes, land mine clearance, and the perform- 
ance of heat-homing devices. 

An important adjunct of the probability and 
statistical work of the Panel was a statistical con- 
sulting service for various Army, Navy, and NDRC 
agencies. Although some of this consulting was done 
in connection with formal AMP studies and projects 
in such a way that the results are adequately reported 
in original Panel reports or the Panel’s Summary 
Technical Report, a large fraction of it was informal 
and the results of it are to be found in reports and 
memoranda of many agencies, particularly Divisions 
2, 5, 8, and 11 of NDRC; Joint Army-Navy Target 
Group, Army Air Forces Board; Proving Ground 


Command, Eglin Field, AAF; Operational Analysis 
Division, Twentieth Air Force, AAF; Combat An- 
alysis Unit, Statistical Control, AAF; Office of the 
Quartermaster General; Navy Air Intelligence 
Group; Navy Operational Research Group; and 
the Guided Missile Committee of the Joint Chiefs 
of Staff. 

Men from several of the Panel’s research groups 
acted as consultants to these various agencies for 
periods ranging from two months to two years. In my 
opinion some of the most useful service which the 
Panel was able to render came about through the 
work of these men in their capacities as consultants; 
the effectiveness of this work increased constantly 
until the end of the war. The work of these men varied 
widely: assistance in setting up sampling inspection 
plans for procurement of materiel, helping in the in- 
troduction of a quality control system in rocket pro- 
duction, working on designs of experiments for toxic 
gas bombing, testing controlled missiles, cooperation 
in the preparation of an incendiary manual, and 
dozens of other projects. 

I cannot leave the topic of mathematical statistics 
without emphasizing the powerful yet severely prac- 
tical role which this relatively young branch of ap- 
plied mathematics has played in the work of the 
Panel. The tools of the probabilitist and statistician 
have been applied to an almost unbelievably wide 
array of problems. Probability analysis played a 
fundamental part in a priori investigation of various 
kinds of weapons and tactics studied by the Panel. As 
the war progressed and these weapons and tactics were 
tested at the proving ground and tried out in com- 
bat, the analysis of the observational data became 
primarily statistical. The work of the Panel surely 
indicates that the Army and Navy will do well in 
their research, development, and testing of weapons 
and tactics to see to it that the tools of the mathe- 
matical statistician are not overlooked. 



PART I 


BOMBING STUDIES 




Chapter 1 

GENERAL CONSIDERATIONS 


11 INTRODUCTION 

I N CHAPTERS 1 to 5 are presented the principal re- 
sults of probability and statistical studies of vari- 
ous bombing problems which have been carried out 
by the Applied Mathematics Panel [AMP] . It has 
proved difficult to choose an order of presentation 
which is consistent and logical, for there does not 
appear to be a completely natural order. The order 
finally chosen is a somewhat artificial one suggested 
by operating conditions in World War II. For exam- 
ple, if the basic assumptions underlying an investiga- 
tion are such that they are at present most nearly 
realized in single-release bombing, the topic is dis- 
cussed under that heading, and similar groupings of 
investigations are used for train and pattern bomb- 
ing. A weakness of the scheme lies in the fact that 
the assumptions which accurately characterize one 
situation often constitute a useful idealization of 
quite a different operational problem; furthermore, 
in a number of instances the position of a study has 
been assigned almost arbitrarily. In view of this, it 
is recommended that the interested reader scan sec- 
tions in addition to those which are obviously ger- 
mane to his immediate problem. 

12 the bombing problem 

In probability and statistical studies of bombing 
problems the two fundamental constituents are: 

1. The target, which comprises a set of areas. In 
particular, it may be a single area and this may be, 
effectively, a point. The configuration is usually 
given. 

2. The bomb fall, which comprises a set of impact 
areas. In particular, as in the case of the target, it 
may be a single area and this may be, effectively, a 
point. The configuration may or may not be fixed; 
if it is not, there can be any degree of statistical or 
geometrical dependence between the components of 
the bomb fall. 

The first decision in solving a bombing problem in- 
volves the choice of an appropriate probability state- 
ment regarding the relationship between the target 
and the bomb fall. For example, it may be desired to 
know the proportion of the target which will, on the 


average, be blanketed by the bomb fall, or to know 
the proportion of the bomb fall which will be con- 
tained within the target, or to know the probability 
that the target will be hit at least a specified number 
of times, or to know the probability that at least a 
specified number of target elements will be hit. 

Making the best objective choice of the measure 
to use is one of the two truly difficult parts of the 
bombing problem, the other being to design tactics 
which are operationally feasible. Once the statistic 
which will measure success has been selected and the 
general domain of feasible tactics entered, it is simply 
a matter of technique and craftsmanship to arrive at 
some specific answers as expeditiously as possible. In 
the first part of the problem the military man is 
usually extremely weak and in the second part the 
scientist is usually extremely weak; in fact, they need 
each other more than either usually thinks. 

There are many occasions when a thorough appre- 
ciation and knowledge of strategic plans, such as can 
only be supplied by military personnel, is needed in 
order to arrive at the proper formulation of the state- 
ment. For example, a uniquely important target, say 
the von Tirpitz, or a heavy-water plant, would re- 
quire a different approach than would a campaign 
against an industry, say that of synthetic oil, even 
when the crucial installations are the same size as 
that of the unique target. Again, the proper statement 
may change as the result of changes in Air Force 
technique. For example, the early visual bombing 
operations in the European Theater of Operations 
were so ragged that there was little question that 
the average proportion of bomb fall contained with- 
in the target was a reasonable measure of perform- 
ance, and that the performance could stand vast 
improvement. However, toward the end of the war, 
as a consequence of increased bomb loads and im- 
proved technique, there was some question whether 
preoccupation with the average proportion of hits 
was not leading, at least on occasion, to over- 
bombing, and whether one should not replace this 
criterion with a new one, say the number of target 
elements or cells hit. This particular issue is contro- 
versial, but the fact that it is controversial suggests 
that the operations may actually have been near- 
ing a point where a re-evaluation of the criterion was 
in order. 




9 


10 


GENERAL CONSIDERATIONS 


13 DIGRESSION ON TWO STATISTICS 

Although a full discussion of the two statistics 
briefly described here is beyond the scope of the 
present volume, it may not be inappropriate at this 
point to comment briefly on two criteria. In order to 
exemplify some of the considerations which enter the 
bombing problem, let us consider (1) the probability, 
say Pk, of at least k hits on a target, and (2) the 
expected number, say E, of hits on a target. The 
necessity to choose between these two criteria arises 
frequently. 

Denoting by P'k the probability of exactly k hits 
and by n the number of bombs, the two criteria are 
defined respectively, as follows. 

p, = '^p[, ( 1 ) 

i-k 

V 

E (2) 

1=1 

where Pk is the sum of certain Pi’s and E is the 
weighted sum of all the P'’s from i = 1 to i = n, 
the weights being i. Thus, Pk assigns equal impor- 
tance to any number of hits greater than or equal to 
k, whereas E values the hits according to their 
number. 

Hence, it appears that to choose between the 
criteria, one must judge (1) that the situation is such 
that less than k hits has relatively little value and 
that more than k hits has little value in excess of 
that associated with k, in which case Pk is a desirable 
criterion, or (2) that the value of the operation is 
almost proportional to the number of hits, in which 
case .E is a desirable criterion. 

But this ignores the question of reliability of per- 
formance, which is best illustrated b}^ reference to a 
special case, i.e., the comparison of Pi and E, for 
here it is easy to demonstrate the point. Referring 
to equations (I) and (2), one observes that the first 
is, in this instance, the simple sum and the second 
the weighted sum of the same (f = 1, • • •, n). 
Suppose one seeks to maximize one of these expres- 
sions, say the one on the right of equation (2). 
Since the expression on the right of equation (1) will 
not, in general, be simultaneously maximized, it fol- 
lows that the use of the criterion E, which leads to 
the greatest number of hits, in the long run will 
cause an unnecessarily large number of missions to 
be complete failures, since Pi, which asks only that 


there be some hitting, is smaller than it need be. 
Similarly, if Pi is maximized, in the long run the 
total number of hits will be unnecessarily small. 

But the choice between P* and E may not be easy 
to make without further quantitative study. For ex- 
ample, suppose it is quite certain that E is the 
primary interest, but that completely sterile missions, 
i.e., missions in which there is no hitting, are some- 
what undesirable from other viewpoints, such as 
morale. It may occur that the conditions which 
maximize E are very unfavorable for P/,, whereas 
those which maximize P;. are very nearly as good 
from the viewpoint of E as are the optimum con- 
ditions for the latter. In the train-bombing case illus- 
trated by Figure 16 of Chapter 3, the use of the 
maximum-P criterion will reduce Pi from a possible 
0.110 to 0.015. On the other hand, if Pi is maximized 
instead of E the cases of some hitting are increased 
more than sevenfold, while E falls below its max- 
imum by but a few per cent. In this case the use of 
Pi seems preferable to the use of E, even though the 
long-term number of hits is highly valued. 

The criteria, P* and P, are not the only possible 
ones of course. The problem of ship-sinking prob- 
abilities, discussed in Section 3.7 under Method of 
Analysis, illustrates another alternative. The ideal 
criterion, when the choice revolves about Pk and P, 
probably is a weighted sum of all Pi’s, namely, 

n 

Jc.p;, (3) 

i=0 

in which the Ci’s reflect accurately the value of i 
hits, including Co < 0 which measures the disadvan- 
tage of complete failure. 

14 THE MATHEMATICAL MODEL FOR 
CALCULATION 

Once the appropriate probability statement has 
been selected, the next step is to invent the mathe- 
matical model, or idealization, which will be em- 
ployed for calculation. Here the desire to introduce 
as much realism as possible must be tempered by the 
knowledge that, unless complexities are built in with 
care, they will cause the labor of calculation to 
balloon seriously. Generally, increasing the accuracy 
of a model is more costly than increasing the pre- 
cision of calculation, and of course the latter is not 
inexpensive. There are, unfortunately, situations in 
which certain of the probability expressions must be 




THE AIMING-ERROR DISTRIBUTION 


11 


determined quite realistically and within close limits, 
for example, when the quantities in question must be 
raised to high powers to obtain the final expression. 

After the model is chosen, it is usually necessary to 
explore, via calculation, a substantial region in the 
parameter space in order to gain a sufficient and use- 
ful appreciation of the behavior of the function. The 
purpose of this exploration is twofold: (1) to discover 
whether the results depend sensitively on the pa- 
rameters of the model, and (2) to discover which of 
the operationally controllable factors can most profit- 
ably be modified, and to forecast the value of attain- 
able modifications; alternatively, if the situation is 
completely new, its probable worth compared to ex- 
isting situations may be estimated. 

The problem has been stated with considerably 
more generality than is useful in the applications, for 
even when the type of probability statement is 
agreed upon, the description of the general target and 
bomb-fall complexes mentioned earlier requires more 
parameters than it is feasible to introduce in a reason- 
able computing program. Since a description of the 
target and bomb-fall complexes, which would include 
both the shapes and positions of the regions, is ex- 
tremely difficult, the problem must be attacked piece- 
meal. Moreover, one must become reconciled to solving 
only those cases to which the greatest interest attaches, 
for even the less general cases are often tedious. 

The target and bomb-fall complexes treated in the 
various sections of the following chapters are special- 
ized in so many different ways that few compre- 
hensive observations can be made regarding them. 
The same is true of the probability statements in- 
corporated in the various studies; they are too varied, 
in accordance with the needs of the immediate prob- 
lems attacked, to fit into a summing-up statement. 

THE AIMING-ERROR DISTRIBUTION 

One element, however, is common to the great 
majority of the investigations and warrants some 
discussion. This element is the distribution of aiming 
errors. The distance from the intended mean point 
of impact [MPI] to the centroid of each independ- 
ently aimed bomb fall is assumed to be a random 
vector from a two-dimensional Gaussian distribu- 
tion. The dispersion of aim may or may not be the 
same in range as in deflection; the components may 
or may not be correlated; the intended MPI, or long- 
term average position, may or may not be at target 
center. This much latitude and variation occurs, but 


otherwise the assumption regarding aiming-error dis- 
tribution is fixed. Practically every investigation in- 
tended for direct application to bombing is based on 
the assumption that the aiming-error distribution is 
Gaussian, or normal. It is pertinent and important, 
therefore, to comment on this assumption. 

There is abundant evidence from practically every 
type of bombing operation — practice and combat, 
single-release and pattern, conventional bomb and 
controlled missile, visual and radar sighting — which 
shows that bombing errors are not normally distrib- 
uted and, in fact, from some viewpoints, that the 
Gaussian is not a particularly good approximation to 
the actual distribution. 

Empirical bombing distributions usually have too 
many large deviations, measured from the mean, 
compared to normal distributions. These empirical 
distributions generally can be represented as the sum 
of two component distributions, of which at least 
one is normal, and this normal component usually 
comprises the major portion of the data. In fact, 
usually three-fourths or more of the observations fall 
in this category. The other component distribution, 
which, since it is often weakly represented and al- 
ways unwelcome, may be called the contaminating 
distribution, plays a role of varying importance de- 
pending on whether the investigation is largely a 
priori or largely a posteriori. Some of the consequences 
of each type of investigation will be examined. 

In many bombing problems it is of vital impor- 
tance that the probability density in the neighbor- 
hood of the intended MPI, usually at target center, 
be reliably estimated, since the important target is 
often small and situated there. Suppose that, pre- 
dominantly, the aiming errors, x and y, in range and 
deflection, respectively, may be represented by the 
circular-Gaussian density function centered at the 
intended MPI 

p = _l_g-(l/2.=)to^+rt , (4) 

2Tr(T^ 

where a is the standard deviation of aiming error in 
range or deflection, but that a fraction, say a, of the 
aimings are associated with some other density func- 
tion. For the sake of an explicit example, let this 
too be a normal density function, centered at the 
intended MPI with standard deviation, a' = ra. Then 
the true density in the neighborhood of the origin is 




12 


GENERAL CONSIDERATIONS 


Suppose that an a priori investigation is under- 
taken and that the presence of the dominant distri- 
bution equation (4) is recognized and its standard 
deviation estimated to be da, but that the contamina- 
tion is not suspected, the estimated density might be 


Now consider an a posteriori investigation in 
which one is presented with data, from s actual 
bombing operations, subject to the density law 
equation (4) plus a contamination term. Suppose 
again the existence of the contamination is not rec- 
ognized and that one is willing to assume that the 
data are subject to a circular-Gaussian law. Suppose 
that the variance, o-^, is estimated from the s ob- 
servations by efficient processes, e.g., by 

r s s 

[2 ~ 1 J ’ 

where x and y are the sample averages of the co- 
ordinates X and y. The expected value of this statistic, 

Eia^) = [l — q: + , (8) 

indicates that the density estimate would be ap- 
proximately 

Taking the ratios of estimated to true density for the 
hypothetical a priori and a posteriori cases, i.e., 
equation (6) and equations (5) to (9), one finds 



Even if the amount of contamination a and the rela- 
tive magnitude r of its standard deviation are modest, 
Pe may be in error by a substantial factor. For exam- 
ple, with q; = 0.1 and r = 5, p^/p = 0.325, i.e., 
Pf, is in defect by a factor greater than 3. 

The situation is illustrated in Figure 1 for a ran- 
dom sample of s = 20 items drawn from a circular- 
Gaussian population with standard error a; the 
arrows indicate the shifts which would affect the 
last two items drawn (a = 0.1) if they were from 
a contaminating distribution in which a' = 5cr. 

The moral of the example is twofold: (1) that an 
a priori investigation is not seriously prejudiced by 
virtue of being based on the assumption of normality 


when, in fact, the distribution is contaminated, and 
(2) that, in a posteriori evaluation of bombing data, 
an uncritical acceptance of the hypothesis of nor- 
mality may result in very and unnecessarily poor 
estimates of aiming errors. 

The question as to how such data may best be 
handled deserves more study than it has received to 
date. The task of decomposing an observed distri- 



Figure 1. A random sample of 20 observations from 
a circular-Gaussian distribution with standard devia- 
tions of a. The translations, indicated by arrows, show 
the change which takes place when the last two observa- 
tions are drawn from a Gaussian distribution with stand- 
ard deviation 5a. 

bution into component distributions, and of making 
unbiased, efficient estimates of their parameters, 
offers difficulties. It is, moreover, unsatisfying labor 
when the sample is small, as is so frequently the case. 
A technique which is rapid, and leads to unbiased 
estimates of probability, but is quite inefficient, is 
the following: Superpose the bomb fall on the target 
T of interest and count the hits. If there are H of 
these then the ratio p = // /s is an unbiased estimate 
of the probability of hitting. Now assume that the 
distribution is normal and set 

T 

On solving for a one obtains a value which can be 
used safely for calculating probabilities of hitting 
targets which do not differ radically from the T used 
in equation (11). But as mentioned above, the reli- 
ability of these estimates is often not as great as 
could be desired from a sample of s observations. 

TERMINOLOGY AND NOTATION 

The results summarized in the following chapters 
of Part I have been worked out over a period of 




TERMINOLOGY AND NOTATION 


13 


several years by a large number of individuals. The 
terminology and notation which appear in the orig- 
inal papers are quite naturally not consistent when 
these papers are viewed as a whole. Moreover, they 
are sometimes made cumbersome by the need to re- 
flect the details of an argument, which are unneces- 
sarily fine for a summary report such as this. There- 
fore an attempt is made, if only partly successful, to 
make the terminology and notation used in the fol- 
lowing chapters simple and consistent to the extent 
that it is feasible to do so with reasonable labor, in 
order that the presentation may be largely self- 
contained, relatively easy to read, and not unneces- 
sarily burdened by unending lists of definitions. 

The following short list of symbols is not compre- 
hensive, but does represent the general intention of 
the writer. A knowledge of them at the outset will 
probably make the reading easier. 

n Number of bombs released by an aircraft 
during one bombing run; in other contexts, 
the number of sections or cells in a target. 

N Number of bombs released by a formation 
of aircraft during one bombing run; in other 
contexts, the total number of bombs con- 
sidered. 

s Number of attacks per target, by single 
aircraft or by formations of aircraft. 

s' Number of aircraft dispatched per target. 

ffa Standard deviation of the aiming-error 
distribution in one dimension, either range 
or deflection. In case the standard deviations 
in range and deflection are unequal, the 
symbol aa is replaced by aar and aad, indi- 
cating the two components in range and de- 
flection, respectively. 

(Td Standard deviation of the bomb-disper- 
sion distribution in one dimension, either 
range or deflection. In this presentation the 
components are immaterial; e.g., the range 
component in a lengthwise attack on a very 
long target. 

MRE The mean radial aiming error (MRE = 
1.2533o-afora circular-Gaussian distribution). 

CEP The so-called circular probable error; ac- 
tually the median radial error (CEP — 
1.1772o-„fora circular-Gaussian distribution). 

Pk The probability of at least k hits in a 
single attack. 

P'k The probability of exactly k hits in a 
single attack. 


Pics The probability of at least k hits in s 
independent attacks. 

P'lcs The probability of exactly k hits in s 
independent attacks. 

E The expected number (or proportion), or 
long-term average number, of hits. 

I Intended spacing usually of bombs in 
train, i. e., the spacing which would be ob- 
tained in train, if ad were zero. 

A circumflex (-) over a symbol connotes the best 
value of the quantity discussed, from a viewpoint 
which is clear from the context. Thus I may be the 
value of I which maximizes Pk] this maximum may 
be indicated by P/;. The circumflex is occasionally 
used to connote an estimate, e.g., a"^ as an estimate 
of (7^, which is a departure from the rule unless one 
is willing to consider best estimates as part of it. 

The statistics MRE and CEP are used more often 
than a a by Service personnel, for these can be calcu- 
lated from operational data by somewhat simpler 
formulas than that needed for the calculation of Ca. 
However, the latter statistic is much more convenient 
for theoretical studies for much the same reason that 
MRE and CEP are preferred in practice — its use 
leads to less cumbersome theoretical formulas. It has 
certain other advantages, even in the practical field, 
when it is important to derive from the data as much 
information as possible (which is usually desirable, 
of course), but uncritical use of a a is more apt to 
give trouble than use of the simpler statistics. For 
example, an estimate of CEP obtained by counting 
is very little affected by contaminated data. This 
fact probably more than offsets the disadvantage of 
using an inefficient statistic. 

It is of course not important that the quantities 
used to describe the geometry of a problem — the 
standard deviations of the aiming errors and of bomb 
dispersion, aa and ad, the dimensions of target and 
bomb fall — be expressed in feet, for only the ratios 
of these quantities to any one of their number are 
important. Thus a train-bombing problem involving 
a 100 X 600-ft target, with aa = 500 ft, ad = 50 ft, 
and I = 140 ft, is essentially the same problem as one 
characterized by T = 1 X b, aa = 5, ad = 0.5, and 
I = 1.4. For this reason the unit of measurement can 
be left unspecified, and it is usually desirable to 
do so (1) because the results then flaunt the little 
generality to which they are entitled, and (2) because 
a large number of zeros are eliminated from text 
and tables. Usually, hut not invariably, the target width 
or the aiming-error statistic is taken as the unit. 


Chapter 2 

SINGLE-RELEASE BOMBING 


2 1 INTRODUCTION 

S ingle-release bombing has played a very small 
role in the level- bombing operations of World 
War II. Generally, the level bombers — light, me- 
dium, heavy and very heavy — have been eager, with 
good reason, to expend their loads on one bombing 
run, and usually in a formation release. 

This inability to use single-release methods, al- 
though mildly inefficient, probably has not resulted 
in a catastrophic loss in hitting power, measured by 
the average number of hits on the target area, often 
2,000 ft or more on a side. The above view is based 
on two items: (1) an hypothesis that single-release 
bombardiers’ aiming errors would not give on the 
average a pattern which is much better, if any, than 
those achieved for patterns by lead bombardiers, who 
presumably are superior combat personnel, and (2) 
the fact that once the pattern is reduced to a size 
comparable to that of the target, the expected num- 
ber of hits is relatively insensitive to further reduc- 
tions, and then single-release bombing is simply a 
limiting case in which the pattern is zero. 

In the war, single-release bombing has been almost 
exclusively the forte of the dive bomber, the fighter 
bomber, and the torpedo bomber. It is a wry situ- 
ation which results in the single-release bombing be- 
ing done exclusively by aircraft and tactics in which 
the sighting problem is solved crudely or with great 
difficulty, while the level bombers, with fine visual 
sights designed exclusively for single-release bomb- 
ing, do none of it. 

The AMP has had little contact with the single- 
release type of operation during the war and there- 
fore has not analyzed some of the more interesting 
and difficult situations, such as the attack of ma- 
neuvering targets by carrier-based aircraft. Conse- 
quently, the studies made of single-release bombing 
have usually been rather special in character. Items 
such as guided-missile bombing and air-to-air bomb- 
ing are the principal ones on the list. 

However, despite the restricted application of 
single-release bombing, the theory of such bombing 
is often a good first approximation to more complex 
bombing operations. This theory, although some- 
times a little onerous computationally, is so much 
simpler than that required for a good treatment of 


the complex operation that it is profitable to use it 
for preliminary exploration and for the development 
of useful mnemonics. Therefore, additional space is 
devoted to it here. 


2 2 FORMULAS AND APPROXIMATIONS 

The probability of hitting a target of area T with 
a single bomb when the aiming errors in range and 
deflection are independently and normally distrib- 
uted is 

T 

where (Tar a-nd (Tad are the standard deviations of 
the aiming error distribution in range and in deflec- 
tion, respectively. The integration is conducted over 
the area of the target. 

It is often difficult to use equation (1) because, in 
general, it requires numerical integration. The four 
exceptional target areas for which the necessary 
functions have been tabled are: 

1. Rectangular targets. P may be computed using 
well-known single- and bi-variate normal probability 
tables. 

2. Right-triangle targets, with the aiming point at 
an acute vertex. P has been tabulated by AMP.^’^ 

3. Circular targets. P has been tabulated by AMP"^ 
for the case (Tar = When the center is the aim- 
ing point, the integration can be performed ex- 
plicitly. 

4. Elliptical targets, with the aiming point at the 
center and (Tar = <Tad. P is expressible in terms of the 
P for circular targets, which is tabled, as mentioned 
above. ^ 

These are the principal cases in which numerical 
integration may be avoided. It would obviously be 
desirable to have at hand a rapid method for com- 
puting P for regions of any shape and with the aim- 
ing point anywhere. A numerical method, usable 
when (Tor = (Tad = (To, is to evaluate 

P = ^ f (1 - C (2) 

Jc 


APPROXIMATIONS TO P 


15 


integrating round the contour c, say (f){r,d) = 0, 
of the target. This process is expedited by use of an 
inverse table of the integrand which has been pre- 
pared by AMP.^ 

In single-release bombing the quantities aar and 
(Tad include the components of variance associated 
with bomb-dispersion which must be explicitly 
accounted for in more complex operations, such as 
in train bombing. 

There are several uses for equation (1). As stated, 
it is an exact expression (the assumptions being 
granted) for the probability of hitting with one 
bomb. The quantity sP is an exact expression for the 
expected number, E, of hits in a series of s single 
releases. The quantity nP is exact for the expected 
number E of hits in a scatter bombing attack, pro- 
vided one uses standard deviations, say a'ar and 
(Tad, defined by 


t2 _ 2 , 2 

^ar ^ar i (^d 

^'2 _ ^2 2 
^ad ^ad i O^d , 


(3) 


in which the aiming-error and bomb-dispersion stand- 
ard deviations are suitably combined. It is an approx- 
imate expression for the expected number, Ejn or 
E I N,oi hits per bomb in train and pattern bombing, 
useful when the dimensions of train or pattern are 
small compared to the aiming-error parameter, (Ta. 
In fact, it is exact for pattern bombing when the 
bombs in pattern are normally distributed. 


APPROXIMATIONS TO P 


There are useful approximations to equation (1). 
When the greatest dimension of the target is small 
compared to (Tar and (Tad, equation (1) yields 


P = 


T 

) 

2'7r(7 ar(T ad 


(4) 


where T is the area of the target. 

If (Ta = (Tar = ad, and if One makes use of the 
relationship between da and the mean radial error, 
MRE, in a circular-Gaussian distribution, namely 

MRE = -J I , (5) 


equation (4) may be written 

P- T’. 

^{MREY 4 ’ 


(6) 


i.e., for small targets the probability of hitting is 
approximately one-fourth the area of the target, pro- 
vided the target dimensions are expressed in terms of 


the mean radial error as unit. For example, the prob- 
ability of hitting the target shown in Figure 1 is 
approximately P = 3/(4 X 8 X 8) = 0.0117, which 




-f- 0.473 


+ 

0.495 



Figure 2. Illustration of the dependence of prob- 
ability of hitting on shape of target. All of the above 
targets have the same area, namely the area enclosed by 
the CEP circle. 

compares favorably with 0.0116 obtained by equa- 
tion (1), when the sides at top and right are of unit 
length and when MRE = 8, and where the aiming 
point A P is at the corner indicated. 

Of course, a formula which takes no account of 
target shape must be used only for small targets, for 
when the probabilities are substantial, they do de- 


16 


SINGLE-RELEASE BOMBING 


pend sensibty on the target’s shape. This is apparent 
in the targets of Figure 2, which all have the same 
area. 

The judgment of when equation (6) may be used 
with safety is assisted by reference to another approx- 
imate expression which may be used for rectangular 
targets (T = L X TF) of any size, namel}^ 

TF'2 -ii 

(1-c M(l-e M J . (7) 

This expression is never wrong by more than one 
or two per cent if the aiming point is at the center. 
When equation (7) yields values closely approxi- 
mating those obtained from equation (6), there is 
then no question regarding the applicability of equa- 
tion (6). 

As an example of the accuracy of equations (6) 
and (7), consider a 0.2 X 2 target and MRE = 5. 
Here P = 0.003959 and equations (6) and (7) err, 
in excess, by 4 X 10“^ and 10~®, respectively. 

It is generally appreciated, and evident from the 
equations, that the probability depends sensitively 
on the aiming-error parameter, a a, varying inversely 
with the square of era when the target is small. It is 
interesting to see exactly how this dependence varies 
with the target dimensions. A particular way of dis- 
playing the effect is to answer the question : What is 



l/e 1/4 1/2 1 2 4 8 16 

o-^/W 

Figure 3. The advantage factor A vs cralW . A refers to 
the case when the standard deviation of the aiming 
errors is reduced from o-o to o-a/2. The probabilities writ- 
ten along the curves are based on the value a a. W is 
the target width. 

the advantage factor, A (i.e., ratio of new to old 
probability), when the aiming error is halved? 

Figure 3 shows a plot of A versus (TajW (this is the 
original or unhalved value of (7a), for I X 1, 1 X 6, 
and I X 00 targets. The values along the curves are 
the original probabilities. It is clear that the four- 
fold advantage does not accrue unless <Ja is quite 
large compared to target dimensions. Indeed, in the 


most favorable case, that of the square target, the 
advantage does not approach this value until a a = 4, 
for the 1X6 target (7a must be greater than 8. 
For the 1 X oo target the maximum value of A is, 
of course, 2. 



1/64 1/16 1/4 I 4 16 64 256 



Figure 4. The advantage factor A vs ( 1/s) (aa/W)^. 
A refers to the case when the standard deviation of the 
aiming errors is reduced from <Ta to <Ta/2. W is target 
width, and s is the number of independent attacks. 
Strictlj^, this plot should be a scatter chart, but on this 
scale the computed points are usually indistinguishable 
from the curve. 


Figure 4 shows the advantage factor, when the 
value of (7a is halved, for a square target subjected 
to independent attacks; the probability of at least 
one hit. 

Pi = 1 - (1 - py, (8) 

is the criterion. The plot is of A versus (l/s)(aalWy. 
Strictly, this relationship yields a scatter chart, but 
the calculated points (for s = 1,10,100) are so close 
to a simple curve that only the empirical locus is 
shown in the figure. 


2 4 GRAPHICAL ESTIMATION OF P 

The need for a rapid method of estimating P in 
equation (1), for targets of any shape and for any 
choice of aiming point, was remarked upon earlier 
in this chapter. This need is met, in large part, by 
a graph paper designed by AMP, which is shown 
in Figure 5. This is a cell-diagram representation of 
the circular-Gaussian distribution. Each cell marks 
off a region in which the probability is 0.001, except 
for certain of the outer rings where the cells cor- 
respond to probabilities of 0.00025 and 0.0001, as 
indicated. The dots are at the cell medians. 

To use the cell diagram, one first expresses the 
target’s dimensions in units of aa and then draws 
the target on the diagram (or on a transparent over- 
lay) in such a manner that the aiming point falls af 


GRAPHICAL ESTIMATION OF P 


17 



linear probable error 
Ti iTi i 1 iTi 1 1 it. 


LINEAR PfiOBABILlTV 


LINEAR AVERAGE DEVIATION 


standard deviation 


the center of the cell system. The cell area lying 
within the target contour is then estimated, either 
by counting the dots enclosed, or by counting the 
cells totally enclosed and adding on estimates for the 
cells partially enclosed. Except when the outer rings 
are involved, P is the cell count divided by 1,000. 


The cell-diagram probability paper has been manu- 
factured in two sizes, (o-a = 1 in. and (Ta = 2 in.) 
principally for use in the AMP. However, operations 
analysts have been supplied on request and the 
Bureau of Aeronautics has prepared transparencies 
based on the paper. 


CELLS OF EQUAL PROBABILITY 

(for a circular GAUSSIAN DISTRIBUTION) 


Figure 5. AMP cell diagram for the circular-Gaussian distribution. 



18 


SINGLE-RFXEASE BOMBING 


2 5 APPLICATION OF THE SLIDE RULE FOR 
SMALL-TARGET BOMBING 
PROBABILITIES 

A slide rule, titled Small-Target Bombing Prob- 
abilities, is available for calculation of the expected 
number E of hits and the probabilities Pk of at least 
k hits (k = 1, 2, 3, 4) for attacks on small targets. 

The important assumption made in designing the 
slide rule is: That in a region around the origin, of 
radius about half that of the circular probable error 
CEP circle, the distribution is statistically uniform, 
the density being equal to the central density of a 



Figure 6. The AMP Small-Target Bombing Prob- 
abilities Calculator. 


circular-Gaussian distribution. The answers of the 
slide rule are correct for a small centrally located 
target when the aiming-error statistic, CEP, is esti- 
mated on the assumption that the distribution is 
strictly Gaussian. 

The slide rule, in effect, computes the probability 
that (1) a bomb will fall in this central region, and 
(2) it will hit a target in that region. The rule is 
shown in Figure 6. 

The slide rule for small-target bombing probabili- 
ties has been manufactured in small quantities and 
distributed to some operations analysts and other 
personnel in the Services. 


2 5 ESTIMATION OF CEP FROM STANDARD 
DEVIATIONS 

The relationship between CEP and o' a, 

CEP = aV 2 log. 2 = 1.1772<Ta , (9) 

is widely used to calculate the radius, R = CEP, of 
the 50 per cent circle in a circular-Gaussian distribu- 
tion. Certain Army manuals advocate a formula 
equivalent to 

CEP = ^ 2 (Tar (Tad loge 2 (10) 

for use when the standard deviations in range and 
deflection are not equal. The question arises: How 
may one approximate the radius R of any percent- 
age circle, say p, and can something better than 
equation (10) be offered for the radius of the p = 0.5 
circle? 

The three formulas, 

Rl = i/ 2 (Tar (Tad logel/(l-p) , 

R 2 = ((Tar + (Tad) / i loge 1/ (1 - p) , (H) 

= / ((rlr + O-L) loge l/(l-p) , 
have been compared for 0.1 ^ p ^ 0.9 and 0.5 ^ 
(Tar I (Tad ^ 1- The result is that Ri gives the closest 
approximation when 0.1 ^ p ^ 0.3, R 2 when 
0.4 ^ p ^ 0.75, and Rz when 0.8 ^ p ^ 0.9. R 2 is 
the best overall approximation. Hence, for the radius 
of the 50 per cent circle the formula 

CEP = i<Tar + <Tad) G log. 2 = 0.5887((r<., + aa,) 

( 12 ) 

is recommended; it is more accurate than equation 
(10) and simpler to compute. 

The tables of functions and other items cited in 
the text are contained in several documents pre- 
pared by AMP. ^ 

2 7 SELECTION OF AIMING POINTS FOR 
IMPROVEMENT OF TARGET COVERAGE 

When the target is large relative to the aiming 
errors, it may be desirable to use more than one 
aiming point. The study discussed here^ is concerned 
with this question. Since the mathematics used is 
preeminently that of single-release theory, it is in- 
cluded in this chapter; however, extension of the 
theory to certain classes of pattern bombing is 
justified. 

Purposes of the Study. The purposes of the study are 
(1) to develop methods for estimating the number 
and position of aiming points which will maximize 



IMPROVEMENT OF TARGET COVERAGE 


19 


the expected coverage of the target by the lethal 
areas associated with the bombs, and (2) to estimate 
the number of bombs required to achieve a specified 
expected value of the coverage in optimum attacks, 
in the above sense. 

Method of Analysis. In brief, the method of an- 
alysis is to solve accurately an idealized problem in 
one dimension, i.e., the case in which bombs fall 
exactly on a line segment, and then to apply the 
criteria so discovered to each of two mutually per- 
pendicular cross sections of an area target. This de- 
vice avoids the intrinsic mathematical complexities 
of the two-dimensional case and, while the results 
therefore cannot be strictly accurate for that case, 
there is every reason to believe that they are excel- 
lent for practical purposes. 

This belief is due in part to the fact that the re- 
sults in the one-dimensional case suggest that sub- 
stantial departures from the best spacing of aiming 
points I do not seriously affect the expected fraction 
F of the target covered. This is illustrated in Figure 
7 for the case of two aiming points and a line target 
of length 6 ((Ta = 1). The family parameter is C, 
called the potential coverage, which is the number 
of times the target area T is contained in the sum 
of the lethal areas for all the bombs, i.e., C = sa/T, 
where a = lethal area for a single bomb and s = 
number of bombs. 

Results. The results of the study give the optimum 
number n of aiming points for line targets of various 







0 

01 

o 

















C* 1.6 



















C*0.8 





















C=0.3 























1.8 Z.Z 2.6 3^0 3.4 3^ 


I 


Figure 7. Illustrating the relatively weak depend- 
ence of proportion F of target covered on the spacing I 
between aiming points. In this case there are two aiming 
points and the target is six units long. C = sa/T, the po- 
tential coverage, where s = number of bombs, a = ef- 
fective area of each bomb, T = area of target. 



0.1 .15 .2 .3 .4 .5 .6 .7 ,8 .9 I 1.5 2 3 4 5 6 7 8 9 10 

C 

Figure 8. Best distance e from ends of target to aiming points (intermediate points spaced at / = 2) vs potential 
coverage C = sa/T. s = number of bombs, a = effective area of bombs, T = area of target. 


20 


SINGLE-RELEASE BOMBING 



Figure 9. Expected proportion F of target covered, when optimum number of aiming points are used, potential cover- 
age C = sa/T. s = number of bombs, a = effective area of bombs, T = area of target. 


lengths, T, according to the following scheme, in 
which (Ta = 1 is the unit: 

T ^ 6 n = 2 

n=3 

8 < r n = 1 - e + r/2. 

The last expression is rounded to the nearest integer. 
The symbol e connotes the distance from an end of 
the target to the nearest aiming point ; this is deter- 
mined with sufficient accuracy by Figure 8. 

As a simple rule-of-thumb, the spacing 7 = 2 is 
recommended. 

The expected fraction F of target covered is shown 
in Figure 9 as a function of the potential coverage C, 
and hence as a function of the number of bombs s. 

A detailed discussion of the theory and several 
worked examples, showing how to apply it to line 
and area targets, are given in an AMP paper. ^ 

2 8 SELECTION OF AIMING POINTS FOR 
IMITATION OF COMBAT ERRORS 

The following study® is of interest in connection 
with proving-ground tests of guided missiles. It is a 
detail in the design of a test program for the missile 
RAZON. 

Purpose of the Study. The purpose is to select a 
single-offset aiming point which will have this prop- 


erty: The radial distribution of errors measured from 
target center, which results from having a proving- 
ground bombardier aim at the offset, should be the 
best approximation to the radial distribution which 
is obtained in combat. The weapon under test will 
thus be subjected to aiming errors whose magnitude 
and frequency are comparable to those which occur 
in the field. 

Method of Analysis. The method of analysis is 
quite straightforward. The distribution of the radial 
errors, r, in combat is assumed to be circular Gaus- 
sian; the density is 


The proving-ground standard deviation, say a' a, 
together with an offset aiming point at the distance 
R, gives rise to the following probability density: 

p'(t) _ -l/2a'^^ [r^+R^- 2 Rr cose] ^0 ^ 

R 2Tr(Ta Jo 

p and p' are calculated and plotted for various values 
of r, for a trial value of R. The calculations are re- 
peated using new values of R until, by trial, a satis- 
factory R is discovered. The process can be refined by 
adapting a mathematical criterion to describe the 
goodness of fit of p' to p. 

Results. The preceding calculations lead to graphs 
of the form shown in Figure 10. Graph A shows 




LATERALLY CONTROLLED MISSILES 


21 


the usual distribution of radial errors in combat and 
in practice; graph B compares the distribution of 
radial errors in combat with those in practice with 
an offset. Although the fit in graph B is far from 






B 





xy 

✓ y 

/ 

/ 





o' 500' 1000’ 1500' 2000' 


Figure 10. A. Distribution of radial aiming errors in 
practice (solid) and combat (broken) bombing, at alti- 
tude of 20,000 ft. B. Same, except that practice aiming 
point is offset 700 ft. The practice and combat mean 
radial errors, MRE, are assumed to be 370 ft and 800 ft, 
respectively. 

excellent, it represents a worthwhile improvement 
over that in graph A. 

In tests where the complexity can be tolerated, the 
use of two or more offset aiming points would facili- 
tate a closer fit. 

Further details and results are given in an AMP 
report.® 

29 LATERALLY CONTROLLED MISSILES 

The following discussion relates to work, some- 
times of a very elementary nature, done during the 
development period of the laterally controlled missile 
AZON. 

Purpose of the Study. The purpose of the study is 
to forecast the probable value of AZON compared 
to that of standard bombs in single-release attacks 
on non-maneuvering and maneuvering targets. 


Method of Analysis. For non-maneuvering targets 
it is a simple matter to compute the probabilities for 
AZON and standard bombs, under various hy- 
potheses regarding the aiming-error distributions. 

The maneuvering target is a different matter. 
Since the study is intended to be exploratory rather 
than definitive, a very simple mathematical model is 
used (see Figure 11). The target is assumed to have 
three alternatives at an instant late in the bombing 
run, namely, it may remain on course, or initiate a 
hard turn to right or left. The probability of a turn 
is Pt. The bombardier may suspect, but cannot 
know, that the target will turn. If he gambles that 
it will, he aims short, at the point marked C; the 
probability that he aims short is Pb. Calculations 
are made covering all values of Pt and Pb in the 
ranges 0.25 ^ Pt ^ 0.75 and 0 ^ Pb ^ 1. The scale 
of the target and its maneuvers are intended to 
simulate a fast destroyer under attack from an alti- 
tude of about 15,000 feet. 

Results. The principal results are shown in Figure 
12, where the probability of hitting, P, is plotted. 



Figure 11. Mods! for maneuvering ship attacked 
when on course CD. If ship continues on course or if it 
turns to left (or right), it will arrive at B or at A (or A') 
when bombs strike. Bombardier aims at C if he forecasts 
a turn, otherwise at D. All distances in terms of target 
width. 

not against Wjaa, but against Wjua, where al = 
(j'a + (IF/2)2. For standard bombs the standard 
deviation of the aiming-error distribution in range 
and deflection is a a', for AZON the range standard 
deviation is Uar = (J'a and the deflection standard 
deviation is Uad = W. 


22 


SINGLE-RELEASE BOMBING 



W/oi, 

Figure 12. Probability P of hitting maneuvering 
target with guided (AZON) an d standard bo mbs vs 
W /(tL] W = target width, aa = <rJ — (TF/2)2, where 
a a is the standard deviation of the aiming-error distribu- 
tions in range and, for standard bombs, in deflection as 
well. The ranges of values indicated by the cross-hatch- 
ing for a fixed value of W jaa correspond to various 
choices of the probabilities that the ship will turn and 
that the bombardier thinks it will turn. 

That there are probability belts, rather than single 
curves in the figure, is due to the fact that probabil- 
ities P are calculated for ranges of values of Pr and 
Pb. The upper boundary of a belt corresponds to the 
most favorable pairs of values of Pr and Pb from 
the viewpoint of the bombardier, i.e., the probability 
is high that the bombardier will successfully antici- 
pate the ship’s maneuver. The lower boundary cor- 
responds to the least favorable pairs. 

2 10 AIR-TO-AIR BOMBING 

The results presented in this section refer to a pre- 
liminary evaluation of the potentialities of air-to-air 
bombing. 

Purpose of the Study. The purpose of the study^’* 
is to estimate the probability of hitting a medium 
bomber {Japanese Betty: Mitsubishi Type 01), two 
tactics and three fuzings being considered. The tac- 
tics are (1) high attack (1,000-3,000 ft) from the 
rear and (2) low frontal attack (300-500 ft). The 
fuzings are (1) percussion fuze, (2) time-plus- 
percussion combination, and (3) proximity fuze. 

Method of Analysis. The analysis is made in terms 
of the two tactics. 

1. High attack from rear. For percussion fuze, the 
target is a plane contour similar to a plan-view of the 
aircraft. For time-plus-percussion fuze, the target is 


assumed to comprise a vertical right-cylinder such 
that the walls and bases are at a distance r from the 
above-mentioned contour, but there is no target di- 
rectly beneath the contour. For proximity fuze, the 
target is that horizontal cross section of the cylinder 
which contains the contour. 

2. Low level frontal attack. For percussion fuze, the 
target is a plane contour similar to a frontal view of 
the aircraft. For time-plus-percussion fuze, it is a 
horizontal right-cylinder such that the walls and bases 
are at a distance r from the contour, and the cylinder 
is hollow immediately to the rear of the contour. 
For proximity fuze, the target is that vertical cross 
section of the cylinder which contains the contour. 

For the high rear attack the aiming errors in range 
and deflection are measured by a a] vertically, in the 
case of the time fuze, by cr„. For the frontal attack 
the bomb trajectory is almost horizontal, so aiming 
errors are measured vertically and horizontally (and 
again characterized by a a)', in the time-fuze case 
there is an error along the line of sight (strictly, 
along the trajectory) measured by, say, which 
reflects fuzing and ranging difficulties. 

The models used for targets and aiming-error dis- 
tributions are the important items in this analysis; 
the rest is arithmetic. 

Results. The results of the study are presented in 
a series of tables similar to Tables 1 and 2. If it is 
desired to compare the tactics, it is suggested that 
nearly equal values for (r„ and ah be chosen, but that 
values of aa, such as 200-400 ft, for the high attack 
will be most realistically compared with a value of 
aa of 25-100 ft for the low attack. The latter values 
imply very good instrumentation, of course, such as 
the angular-rate bombsights may afford. 

Further details and results are contained in two 
working papers by one of the AMP research groups/’® 


Table 1. Probability of hitting a medium bomber in 
a high level rear attack using contact fuze, or using 
proximity fuze which detonates at a distance r. 


a 

Contact 
r = 0 

r = 25 

Proximity 
r = 50 

r = 100 

50 

0.072 

0.454 

0.735 

0.972 

100 

0.021 

0.142 

0.280 

0.581 

200 

0.0045 

0.038 

0.080 

0.198 

400 

0.0011 

0.0096 

0.020 

0.055 


Table 2. Probability of hitting a medium bomber in 

a 

low level frontal attack using contact fuze, 

or using 

proximity fuze which detonates at 

a distance 

r. 


Contact 


Proximity 


<Ta 

r = 0 

r = 25 

r = 50 

r = 100 

25 

0.057 

0.71 

0.96 

1.00 

50 

0.015 

0.31 

0.62 

0.94 

100 

0.0037 

0.096 

0.23 

0.52 


Chapter 3 

TRAIN BOMBING 


31 INTRODUCTION 

ALTHOUGH THE United States entered World 
War II with first-class single-release bombing 
equipment, such as the high-altitude synchronous 
bombsights designed by Norden (M-14) and Sperry 
(S-1), the thought given to the problem of train 
bombing had resulted only in the development of 
certain auxiliary equipment, notably the mechanical 
and electrical intervalometers of the A and B series, 
which had been installed in bombardment aircraft. 
In a strict sense, the resulting combination of bomb- 
sight and intervalometer constituted a makeshift so- 
lution to the train-bombing problem, for only the 
first bomb released, of those released in a train, 
could be aimed with the precision normally associ- 
ated with the synchronous sight. Equipment ex- 
plicitly designed for the train-bombing task would 
have aimed the center of the train with single-bomb 
precision. Actually, it is possible to aim the center of 
the train with this equipment, but only by falsifying 
the input data fed to the bombsight. Unfortunately, 
the bombsight is not completely fooled by this arti- 
fice and consequently the equipment does not, even 
under proving-ground conditions, aim trains as pre- 
cisely as it aims individual bombs. 

Thus our equipment and, incidentally, our train- 
ing program for bombardiers emphasized the single- 
release bombing problem. On the other hand, it ap- 
peared, from the daily practices of the British and 
Germans, that in current warfare against a first-class 
opponent one could not afford to make more than 
one bombing run per sortie and hence that the prob- 
lem of train bombing would be very much to the fore. 
Subsequent experience indicated that this viewpoint 
represented a step in the right direction, but a much 
too modest one, for by the time our daylight bombers 
appeared over Europe the opposition was so strong 
as to preclude anything except formation bombing. 

Before combat experience showed that single- 
release bombing was not going to be the principal 
technique of bombing in the war, the Services and 
the National Defense Research Committee [NDRC] 
initiated work on the various theoretical aspects of the 
train-bombing problem. As a result, a considerable 
body of information on the subject is now in existence. 


As early as 1932, the British had set up the funda- 
mental equations which govern the probabilities in 
train bombing, but had abandoned these in favor of 
approximate formulas which were much more tract- 
able, computationally. As a matter of fact, for aim- 
ing errors of the magnitude produced in British com- 
bat bombing operations, where area bombing was 
quite frankly the goal, the approximations used were 
entirely adequate. But for the level of accuracy 
achieved in practice operations with our single- 
release equipment, even when it was misused to lay 
trains, it appeared that the more precise mathe- 
matical formulation would be needed. Accordingly, 
the work of AMP was based on the computationally 
difficult formulas. 

The theory of single-release bombing is usually 
quite simple; predictions can be based on a single 
distribution function, or probability-density func- 
tion, and this is often of simple form. Train bombing, 
however, requires for its description the use of two 
distribution functions. The complexities of the prob- 
lem are in large part due to the fact that these two 
distributions preserve their individuality to the end; 
not until the number of bombs in train is reduced to 
one do the functions combine nicely. These distribu- 
tion functions are defined below. 

Consider n equally spaced points lying on a line. 
These points mark the intended relative positions of 
the bombs in the train. The centroid of these points, 
the point midway between the first and last, is de- 
fined as the train center. 

The position of the train center relative to the 
target, and the position of a bomb of the train at im- 
pact, relative to the point which marks its intended 
position in the train, are random variables (with two 
components) . The distribution functions which meas- 
ure the probability density of the variables at all 
points in the plane are called the aiming-error distri- 
bution and the dispersion-error distribution. They 
are usually postulated to be two-dimensional Gaus- 
sian distributions. Thus the aiming-error distribu- 
tion has to do with the position of the train consid- 
ered as a single unit, whereas the dispersion-error dis- 
tribution has to do with the behavior of the bombs 
within the train, regardless of where the train has 
fallen. 




23 


24 


TRAIN BOMBING 


3 2 PROBABILITIES OF HITTING 
RECTANGULAR TARGETS WITH 
SINGLE ATTACKS 

The fundamental study of train-bombing proba- 
bilities was initiated by the Ballistic Research Lab- 
oratory [BRL] of the Aberdeen Proving Ground. 
BRL had computed train-bombing probabilities for 
the one-dimensional case, i.e., for very long tar- 
gets, and requested that AMP extend the inquiry 
to the two-dimensional case. 

This proved to be the most arduous computing 
task undertaken by AMP during the war, due to 
the large number of parameters involved and to the 
tediousness of the individual calculations. The calcu- 
lations were done, in part, by hand machines, but 
the greater part was done on IBM punched-card 
equipment. 

The fundamental formulas, and consequently the 
probability results, have application to fields other 
than train bombing; e.g., torpedo spreads, naval 
gunnery, aerial gunnery. 

Purpose of the Study. The purpose of the study is to 
calculate probabilities of hitting with a train of 
bombs under various hypotheses concerning the aim- 
ing-error and dispersion-error distributions, the size 
of the target, the angle at which it is attacked, the 
number of bombs in train, and the spacing of bombs 
in train. The ultimate goal is to ascertain the im- 
portance of these factors and to discover the opti- 
mum values of the controllable inputs, particularly 
the optimum spacing in train, say I. 

That there is such a thing as optimum spacing, in 
general not zero, may be appreciated from qualitative 
considerations. Suppose that the design is to try to 
achieve at least one hit on a long narrow target by 
attacking across it, which is usually the best course. 
If the bomb spacing is very small the train will be 
short in length and it is likely that, because of the 
aiming error of train center, none of the bombs will 
strike close to the target. Now suppose that the 
spacing between adjacent bombs is large: The train 
will be long and there is a much better chance now 
that the target will lie within the train; but the 
large spacing makes it likely that the target, while 
bracketed, will not be hit. Intuition suggests that 
there is some particular value of the spacing, say /, 
which offers the best compromise, and this is indeed 
the case. 

The problem of finding the value of the best 
spacing is too difficult to be solved by intuitive argu- 


ments; in fact it is difficult to guess, usually, whether 
the best spacing is less than or greater than the 
target width. 

Method of Analysis. A more rigorous method of 
analysis was followed. The probability P* of hitting 
a rectangular target T at least k times in a single 
attack with a train of n bombs, may be written as 


where 


and 


Pk 


PaGkdXdY, 


(i-l) 2 

j=k 


Pi = 



PddXidyi , 


( 1 ) 

(2) 

(3) 


{X,Y) and (xi,yi) being coordinate sj^stems, each 
oriented in the directions of range and deflection, 
with their origins at target center and train center, 
respectively. 

Here Gk and pi are the conditional probabilities for 
obtaining at least k hits with the train, and for hitting 
with the fth bomb of the train, respectively, given 
that the aiming error is (X,7). The right-hand sum- 
mation in equation (2) is the overall combinations of 
the n values of p, taken j at a time. 

Also, pa and Pd are the aiming-error and dispersion- 
error densities, both assumed to be two-dimensional 
Gaussian functions. In almost all of the calculations 
it is assumed that these are circular distributions, i.e.. 


n = — — p (-l/2aS)(X2 + 72) /.x 

Pa (4) 

and 

P. = . (5) 

27r(Td 

The angle of attack 6 measured from the long 
axis of the target, and the spacing I between bombs 
in train are contained implicitly in the limits of in- 
tegration, T. 

The form of Pk exhibited in equations (1) to (3) 
is not the one used for calculation. At the expense of 
some algebra it may be thrown into various less con- 
cise forms which, however, are much more useful in 
practice. The question as to the best form for calcu- 
lation is a difficult one which requires and merits 
much time when an extensive computing program is 
undertaken. 

Results. The results of the study are presented 
both in tables and graphs of which Table 1 and Fig- 




PROBABILITIES OF HITTING WITH SINGLE ATTACKS 


25 



X 



I 



I 



I 

I 



I 


Figure 1. Probability of at least /c hits vs spacing I of bombs in train. Target 1 X 6, 0 — 90 , <ro — S, ad — 1- 
I is expressed in target width units. 


ure 1 are typical examples. Here, for fixed values of 
the target dimensions, the angle of attack, the mul- 
tiplicity of hits, the number of bombs in train, the 
aiming-error and dispersion-error distributions, are 
given the probabilities of success as a function of the 
spacing I in train; / is expressed in terms of target 
width as unit. 

Tables and graphs have been prepared for the sets 
of conditions enumerated in Table 2 and may be 
found in AMP documents/’^ 

As anticipated, a characteristic of all this material 
is that there is always an optimum spacing /, such 
that Pk{l) is a maximum, say P*. Usually the curve 
is quite flat in the neighborhood of the maximum, 
so that small changes in the spacing have little effect 
on the probability. However, the maxima tend to 
become more peaked when the number n of bombs 
in train is increased, and when the aiming-error dis- 
tribution becomes more compact, i.e., when the (Xa 
is decreased. 

The maximum probability may occur for any 
value of the spacing, greater or less than target 
width, and including 0. Whether or not the value 
1 = 0 corresponds to the maximum, the curve al- 


ways has a turning point there, i.e., the ^tangent is 
horizontal when 1=0. This qualitative feature was 
not known when most of the curves, which often de- 
pend on relatively few ordinates, were drawn. There- 
fore, values read from curves showing high contact 
with the Pk axis should be discounted in that neigh- 



0 “ 30 “ 60 “ 90 “ 

e 

Figure 2. Maximum prooaoiiity of at least one hit 
vs angle of attack 6, for various values of n, the number 
of bombs in train. Target 1 X 6, tra = 2, o-d = 0.25. 




26 


TRAIN BOMBING 


Table 1. Probability P* of at least k hits vs spacing / of n bombs in train; also maximum probability, P* and optimum 
spacing, I. Target 1 X 6, 0 = 90°, <ra = 8, <rd = 1. 


n 

k: 

1 

2 

3 

4 

5 

n 

k: 

1 

2 

3 

4 

5 



3.45 

0-0.60 




8 

X . 7 

1.80 

0-0.15 

0 

0 

0-0.15 

2 

I \P 

0.028 

0.003 





I \P 

0.091 

0.031 

0.015 

0.006 

0.002 


0.0 

0.025 

0.003 





0.0 

0.060 

0.031 

0.015 

0.006 

0.002 


0.15 

0.025 

0.003 





0.15 

0.062 

0.031 

0.014 

0.005 

0.002 


0.30 

0.025 

0.003 





0.30 

0.066 

0.030 

0.012 

0.004 

0.001 


0.45 







0.45 







0.60 

0.026 

0.003 





0.60 

0.077 

0.027 

0.007 

0.001 

0.000 


0.75 







0.75 







0.90 







0.90 







1.05 







1.05 







1.20 

0.026 

0.002 





1.20 

0.089 

0.018 

0.002 

0.000 



1.35 







1.35 







1.50 







1.50 







1.65 







1.65 







1.80 

0.027 

0.001 





1.80 

0.091 

0.010 

0.000 




1.95 







1.95 







2.10 







2.10 







2.25 







2.25 







2.40 

0.028 

0.001 





2.40 

0.088 

0.005 





2.55 














2.70 














2.85 














3.00 

0.028 





1 O 


1.28 

0.40 

0 

0 

0 


3.15 







I XP 

0.123 

0.047 

0.027 

0.016 

0.008 


3.30 














3.45 







— 







3.60 

0.028 






0.0 

0.15 

0.071 

0.076 

0.043 

0.045 

0.027 

0.026 

0.016 

0.014 

0.008 

0.007 



0.0 

0.15 

0.30 

0.45 

0.60 

0.75 

0.90 

1.05 

1.20 

1.35 

1.50 

1.65 

1.80 

1.95 

2.10 

2.25 

2.40 

2.55 

2.70 

2.85 

3.00 


2.48 0-0.15 0-0.15 
0.052 0.013 0.003 


0 - 

0.000 


0.041 0.013 0.003 

0.042 0.013 0.003 

0.042 0.012 0.002 

0.044 0.011 0.002 


0.048 0.008 0.001 


0.051 0.005 0.000 


0.30 

0.45 

0.60 

0.75 

0.90 

1.05 

1.20 

1.35 

1.50 

1.65 

1.80 

1.95 

2.10 

2.25 

2.40 


0.087 


0.108 


0.046 


0.043 


0.023 


0.010 0.003 


0.013 0.003 0.000 


0.123 0.026 0.003 0.000 


0.118 0.014 0.000 


0.105 0.006 


0.052 0.002 


0.052 


borhood and re-estimated, making use of the infor- 
mation regarding the tangent. 

By plotting the results in various ways, certain 
empirical generalizations may be made: 

1. The maximum probability Pi (i.e., probability 
for best spacing) of at least one hit is greatest when 
the attack is directed across the target, i.e., 6 = 90°. 
This rule is suggested by Figure 2. The rule, how- 




PROBABILITIES OF HITTING WITH SINGLE ATTACKS 


27 


Table 1. Continued 




ever, seems to be an outgrowth of the circumstance 
that the aiming-error distribution is characterized by 
equal standard deviations, da, in range and in de- 
flection. For if these standard deviations, say dar 
and dad, are unequal and if the bomb dispersion is 
negligible {dd = 0), then it can be demonstrated 



0 0.5 1.0 2.0 4.0 8.0 


Figure 3. Optimum spacing / for at least one hit vs 
standard deviation o-a of aiming errors for various values 
of n, the number of bombs in train. Target 1x6, 

6 = 90°, <Td = 0.3. 

that attacks along the target {6 = 0°) yield greater 
values of Pi than attacks across the target (6 = 90°) 
when darl dad > n, and conversely when darj cfad < n, 
where n = the number of bombs in train. 

2. The optimum spacing I of bombs in train in- 
creases when the standard deviation da of the aiming 
errors increases. This is illustrated by Figure 3. The 
dependence of I on the bomb-dispersion distribution 
is less simple. Consider a given set of conditions, in- 
cluding a non-zero value of dd] 1 will have a value 
which may be greater or may be less than the target 
width TF. Now as dd approaches zero, I approaches W. 


Table 2. List of conditions for which the probabilities P* of at least k hits with a train of n bombs have been computed. 
k = 1,2,- • 5; n = 2,4,8,12,16,20. 


Target size 

d 

Oa 

o’d 

I 

1 X 1 

90° 

0.25,0.50,1,2,4 

0.125 

0.0,0.075, • 

• - ,1.650 

1 X 3 

90° 

0.125,0.25,0.50,1,2 

0.0625 

0.0,0.075, • 

• - ,1.200 

1X6 

90° 

1, V2,2,2V2,4,8 

0.3 

0.0,0.2, • • • 

,2.2 


0° 

0.5,1,2 

0.25 

0.0,0.225,0.450,0.675 

1 X 9 

90° 

1,2,4,8,16 

0.5 

0.0,0.15, • • 

-,3.60 


63° 

1,2,4,8,16 

0.5 

0.0,0.1675, 

- - - ,2.3550 


45° 

1,2,4,8,16 

0.5 

0.0,0.2125, 

- - - ,4.2500 


27° 

1,2,4,8,16 

0.5 

0.0,0.335, • 

- - ,6.700 


0° 

1,2,4,8,16 

0.5 

0.0,0.3, • • • 

,6.0 


Note. The probabilities have not been computed for all possible combinations of the parameters. 


28 


TRAIN BOMBING 


3 3 PROBABILITIES OF HITTING 
RECTANGULAR TARGETS WITH MULTIPLE 
ATTACKS 

Heretofore the discussion has concerned the prob- 
ability of obtaining at least k hits as the result of a 
single attack with a train of n bombs. The extension 
to attacks in which more than one aircraft partici- 
pates is now reviewed. It is likely, from qualitative 
considerations, that the probability of hitting, con- 
sidered as a function of the spacing, depends in a 
non-trivial manner on the number of independent 
attacks. 

Purpose of the Study. The present objective is to 
examine this question and, if the situation is as it is 
expected to be, to perform the necessary calculations 
leading to basic tables for multiple attacks against a 


single target. Once these are at hand questions re- 
lating to the attack of several targets may be investi- 
gated. For example, the questions arise: How should 
a given number of aircraft be allocated among sev- 
eral targets? And how many aircraft can be dis- 
patched, economically, against each target? 

The qualitative considerations mentioned above 
are illustrated here. Suppose one wishes to maximize 
the probability of obtaining at least four hits when 
4-bomb trains are used. If only one attack is to be 
made, it is not difficult to guess that the best spacing 
may be 7 = 0. But suppose that the best spacing to 
achieve at least one hit is quite different from 7=0, 
and suppose that the corresponding maximum prob- 
ability Pi is substantially greater than the value of 
Pi for spacing zero. Now, under these conditions, 
consider the problem of obtaining at least four hits 


Table 3. Probabilities of at least k hits when s independent attacks are made, each with a train of 8 bombs spaced at 
the interval 7. Target 1 X 6, 0 = 90°, aa = 2, ad = 0.3. 


k 

7 

s = 1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

1 

0.0 

0.318 

0.535 

0.683 

0.784 

0.853 

0.900 

0.931 

0.953 

0.968 

0.978 


0.2 

0.429 

0.674 

0.813 

0.893 

0.939 

0.965 

0.980 

0.989 

0.994 

0.996 


0.4 

0.577 

0.821 

0.924 

0.968 

0.987 

0.994 

0.998 

0.999 

1.000 

1.000 


0.6 

0.673 

0.893 

0.965 

0.989 

0.996 

0.999 

1.000 

1.000 

1.000 

1.000 


0.8 

(0.706) 

(0.914) 

(0.975) 

(0.993) 

(0.998) 

(0.999) 

(1.000) 

(1.000) 

(1.000) 

(1.000) 


1.0 

0.692 

0.905 

0.971 

0.991 

0.997 

0.999 

1.000 

1.000 

1.000 

1.000 

2 

0.0 

0.258 

0.453 

0.599 

0.708 

0.788 

0.847 

0.889 

0.920 

0.943 

0.959 


0.2 

0.336 

0.568 

0.723 

0.825 

0.890 

0.931 

0.958 

0.974 

0.984 

0.990 


0.4 

(0.414) 

0.683 

0.837 

0.919 

0.961 

0.981 

0.991 

0.996 

0.998 

0.999 


0.6 

0.360 

(0.688) 

(0.864) 

0.945 

0.978 

0.992 

0.997 

0.999 

1.000 

1.000 


0.8 

0.239 

0.639 

0.854 

(0.945) 

(0.980) 

(0.993) 

(0.998) 

(0.999) 

(1.000) 

(1.000) 


1.0 

0.129 

0.558 

0.810 

0.925 

0.972 

0.989 

0.996 

0.999 

1.000 

1.000 

3 

0.0 

0.218 

0.395 

0.536 

0.644 

0.733 

0.799 

0.850 

0.888 

0.919 

0.939 


0.2 

(0.256) 

(0.467) 

0.630 

0.748 

0.831 

0.888 

0.927 

0.953 

0.970 

0.981 


0.4 

0.190 

0.467 

(0.683) 

0.822 

0.905 

0.951 

0.975 

0.987 

0.994 

0.997 


0.6 

0.061 

0.395 

0.673 

(0.840) 

(0.927) 

(0.968) 

(0.987) 

0.995 

0.998 

0.999 


0.8 

0.010 

0.286 

0.602 

0.809 

0.916 

0.966 

0.987 

(0.995) 

(0.998) 

(0.999) 


1.0 

0.001 

0.163 

0.481 

0.730 

0.873 

0.944 

0.977 

0.991 

0.996 

0.999 

4 

0.0 

(0.183) 

0.343 

0.477 

0.586 

0.679 

0.751 

0.809 

0.854 

0.893 

0.916 


0.2 

0.172 

(0.356) 

(0.521) 

0.654 

0.756 

0.831 

0.885 

0.917 

0.948 

0.966 


0.4 

0.041 

0.268 

0.506 

(0.691) 

0.818 

0.897 

0.944 

0.970 

0.984 

0.992 


0.6 

0.002 

0.169 

0.439 

0.672 

(0.825) 

(0.913) 

(0.959) 

(0.981) 

(0.992) 

(0.997) 


0.8 

0.000 

0.066 

0.309 

0.577 

0.774 

0.890 

0.950 

0.979 

0.991 

0.996 


1.0 

0.000 

0.018 

0.169 

0.428 

0.662 

0.821 

0.912 

0.960 

0.982 

0.993 

5 

0.0 

(0.150) 

(0.292) 

(0.421) 

0.527 

0.623 

0.701 

0.764 

0.816 

0.864 

0.889 


0.2 

0.089 

0.240 

0.400 

(0.544) 

0.664 

0.758 

0.828 

0.875 

0.917 

0.944 


0.4 

0.002 

0.136 

0.342 

0.543 

(0.703) 

(0.818) 

0.893 

0.939 

0.966 

0.981 


0.6 

0.000 

0.041 

0.227 

0.466 

0.671 

0.815 

(0.902) 

(0.951) 

(0.976) 

(0.989) 


0.8 

0.000 

0.005 

0.104 

0.321 

0.559 

0.745 

0.866 

0.934 

0.969 

0.986 


1.0 

0.000 

0.000 

0.031 

0.168 

0.387 

0.604 

0.770 

0.877 

0.938 

0.971 


Note. Parentheses mark the probability which is greatest in each column. 


PROBABILITIES OF HITTING WITH MULTIPLE ATTACKS 


29 


Table 4. List of conditions for which the probability Pks of at least k hits in s attacks with a train of n bombs have 
been computed, k = 1,2, • • •, 5; n = 2,4,8,12,16,20. 


Target size 

6 

<Ta 

(Td 

/ 

s 

1 X 1 

90° 

0.25,0.50,1,2,4 

0.125 

0.0,0.075, • • •, 1.650 

1,2, • . . 

, 10 

1 X 3 

90° 

0.125,0.25,0.50,1,2 

0.0625 

0.0,0.075, • • •, 1.200 

1,2, . . . 

,10 

1 X 6 

90° 

1,V2,2,2V2,4,8 

0.3 

0.0,0.2, • • •, 2.2 

1,2, • . . 

,48 


0° 

0.5,1,2 

0.25 

0.0,0.225,0.450,0.675 

1,2, . . . 

,25 

1 X 9 

90° 

1.2,4,8,16 

0.5 

0.0,0.15, • • •, 3.60 

1,2, . . • 

, 10 


63° 

1,2,4,8,16 

0.5 

0.0,0.1675, • • •, 2.3550 

1,2, . • ■ 

, 10 


45° 

1,2,4,8,16 

0.5 

0.0,0.2125, • • •, 4.2500 

1,2, . . • 

, 10 


27° 

1,2,4,8,16 

0.5 

0.0,0.335, • • •, 6.700 

1,2, . . . 

, 10 


0° 

1,2,4,8,16 

0.5 

0.0,0.3, • • •, 6.0 

1,2, . • . 

, 10 


Note. The probabilities have not been computed for all possible combinations of the parameters. 


in, say, four, five, or ten attacks with 4-bomb trains. 
Intuition suggests that it may be more profitable to 
space the bombs in each train so as to try to achieve 
the four hits one by one, rather than to try to get 
all four each time an attack is made. 

Method of Analysis. The method of analysis was to 
let P'lc be the probability of exactly k hits with one 
train of n bombs (as distinct from P* which has been 
used to designate the probability of at least k hits) 
and let Pks be the probability of at least k hits with 
s trains of n bombs. Then 




( 6 ) 


It is easy to show that when A; = 1 the spacing I 
which maximizes Pi also maximizes Pis, but that 
when A: > 1, this is not true. Hence, the character 
of the probability curve, Pksil), depends on s and 
it is necessary to perform additional calculations. 

The questions regarding the best allocation of air- 
craft are investigated by calculating probability av- 
erages with respect to the mission, the bomber life- 
time, etc. Certain of these require quite elaborate 
mathematical descriptions and depend on a number 
of a priori hypotheses, the practical validity of which 
are unknown. However, the importance of this part 
of the study does not depend on the validity of the 
particular hypotheses adopted for discussion; rather, 
it stems from the light that is thus thrown on the 
subject of large-scale bombing, for it is brought out 
that the best bombing policy may depend sensitively 
on the exact formulation of the short- and long-term 
objectives of the Air Forces. 

Results. The results of calculation for multiple at- 
tacks are presented in tables of the form of Table 3; 


these are in fact identical with those for single at- 
tacks, except for the presence of the parameter s. 
Certain of the values in Table 3 are graphed in 
Figure 4. 

Calculations have been made for the various sets 
of conditions itemized in Table 4. Actually this is a 
more restricted calculating program than that for 
single attacks, outlined in Table 2. There are two 



0 0.2 0.4 0.6 O.B 1.0 

1 


Figure 4. Probability Pks of at least'^A hits with s 
trains of n bombs vs spacing I, when k = 4, n = S, target 
1 X 6, 0 = 90°, o-a = 2, (Td = 0.3. 

reasons for this; (1) Many of the questions of primary 
interest could be answered satisfactorily from the 
probabilities at hand, and (2) in the field, the im- 
portance of train bombing had been, by then, com- 
pletely overshadowed by that of pattern bombing, 
and in 1944 AMP curtailed its work on train bombing. 

Figure 5 illustrates the fact that, as anticipated, 
the optimum spacing I depends on the number s of 
attacks as well as on the number n of bombs in train. 
In this example So.go is the number of attacks needed 
to yield a probability of 0.90 that at least 5 hits 
will be made on a 1 X 6 target, when the standard 


30 


TRAIN BOMBING 


deviations, a a and ad, are 4 and 0.3, respectively. 
Figure 6 illustrates the dependence of So.go on the 
values of aa and n. 

Consider the problem of allocating attacks so as to 
maximize the effectiveness (i.e., the number of tar- 
gets hit at least k times) of the attacking force. If the 
force is fixed in size it is evident that this allocation 
will be such that the effectiveness of each attacking 
aircraft is a maximum, which suggests that a new 
table be prepared by dividing by s each of the max- 
imum probabilities Pks in a list like Table 3. Table 5 
has been prepared in this manner. From inspection 
of tables of this type it appears that single attacks 
are most efficient, according to the very simple cri- 
terion used, when the number n of bombs in train 
exceeds the number k of hits required, and that mul- 



O 4 8 12 16 20 


n 

Figure 5. Plot of So .90 vs n, number of bombs in train. 

I is the optimum spacing, and So .90 is the number of at- 
tacks needed to yield a probability of 0.90 that at least 
5 hits will be made on a target. Target 1 X 6, 0 = 90°, 

<Ta = 4, (Td = 0.3. 

tiple attacks are needed when the number n of bombs 
in train is less than or equal to the number k required. 

The last result is not exactly a discovery, but it is 
saved from being trivial by the facts that the opti- 
mum number s of attacks is not the least possible 


and that the yield, Pks/s, is sometimes spectacularly 
greater than that for other, and apparently reason- 
able, values of s. For example, with n = 2 and fc = 5, 
threefold attacks (s = 3) yields P53/3 = 0.00185, 
whereas the use of 21 attacks per target yields 



Figure 6. Plot of so .90 ys <ra, where So .90 is number of 
attacks nedeed to make Pks {k = 5) equal to 0.90 for 
various values of n, the number of bombs in train. 
Target 1 X 6, 0 = 90°, ad = 0.3. 


P 5 21/21 = 0.04010. Thus in this case it is more than 
20 times as effective to allocate 21 aircraft to a 
target as it is to allocate 3 aircraft to each of 7 targets. 

Analyses similar to the above have been made in 
which the optimum number s' of aircraft to dispatch 
to each target, as opposed to the number s to attack, 
is estimated on the basis of various assumptions re- 
garding the loss-rates, and regarding short- and long- 
term values which depend on the replacement rate. 
As a single illustration of these more complex situa- 
tions, consider the problem of determining the num- 
ber s' of aircraft to dispatch to each target in order 
that each aircraft will destroy as many targets as 
possible, say Lks', during its lifetime. Assume that 
the probability that an aircraft will be lost at any 
moment depends linearly on the reciprocal of the 
number of aircraft present, during the combat phase 
of the mission, and that it is zero at other times. The 
expected number of targets destroyed is determined 
from ^ 

^ Os's P ka 



RULES FOR BOMB SPACING 


31 


where C«' is the probability that a specified aircraft 
will survive a mission on which s' were dispatched, 
and Cs’s is the probability that it will be one of the 
s which will survive to the target. Table 6 gives the 
results for a sample computation. 

The multiple-attack tables and related discussions 
are contained in a number of AMP documents.^’^’®"^^ 

34 EMPIRICAL RULES FOR BOMB 
SPACING IN HITTING RECTANGULAR 
TARGETS 

The extensive sets of tables and graphs described 
in the two preceding studies are intended as basic 
reference material. They are not immediately useful, 
usually, for solving specific problems, for their use 
would involve interpolation (often nonlinear) in six 
dimensions. An attempt to use them in this way 


would probably be a frustrating experience, climaxed 
by the discovery that one or another parameter had 
to be extrapolated. 

However, this material is a uniquely valuable 
source of general information for train-bombing 
problems. An important use is for the discovery of 
approximate, but unbiased, rules which may be ap- 
plied to a variety of situations. 

As an example of this, AMP, at the request of 
the Armament Laboratory, Wright Field Proving 
Ground, undertook to develop a calculator which 
would indicate the optimum spacing for bombs 
dropped in train. This work is summarized in the 
following paragraphs. 

Purpose of the Study. The purpose of the study is to 
determine the spacing / for bombs in train, which 
will maximize the probability of achieving at least k 
hits on rectangular targets. 


Table 5. Part of probability Pks/s, of hitting at least k times, ascribable to each bomber of s attacking planes. (All 
values have been multiplied by 100.) Target 1 X 6, 0 = 90°, <ra = 2, <rd - 0.3. 


n 

k 

5 = 1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

2 

1 

31.1 

26.3 

22.4 

19.4 

16.9 

14.9 

13.2 

11.9 

10.7 

9.76 

8.94 


2 

11.0 

11.1 

10.9 

10.6 

10.3 

10.2 

10.0 

9.63 

9.18 

8.70 

8.21 


3 


1.89 

3.31 

4.35 

5.10 

5.64 

6.00 

6.24 

6.48 

6.60 

6.60 


4 


0.604 

1.27 

1.93 

2.54 

3.07 

3.53 

3.90 

4.20 

4.42 

4.57 


5 



0.185 

0.492 

0.867 

1.27 

1.68 

2.07 

2.43 

2.75 

3.02 

4 

1 

51.2 

38.1 

29.5 

23.6 

19.4 

16.4 

14.2 

12.5 

11.1 

10.0 

9.09 


2 

21.1 

20.2 

19.2 

18.4 

16.9 

15.2 

13.6 

12.2 

10.9 

9.92 

9.05 


3 

13.4 

13.2 

12.8 

12.4 

12.3 

12.3 

11.8 

11.2 

10.4 

9.62 

8.89 


4 

6.39 

7.66 

8.41 

8.87 

8.95 

8.99 

9.27 

9.35 

9.17 

8.84 

8.41 


5 


2.26 

3.92 

5.17 

5.92 

6.51 

6.90 

7.24 

7.50 

7.59 

7.55 

8 

1 

70.6 

45.7 

32.5 

24.8 

20.0 

16.7 

14.3 

12.5 

11.1 

10.0 



2 

41.3 

34.4 

25.5 

23.6 

19.6 

16.6 

14.3 

12.5 

11.1 

10.0 



3 

25.6 

23.4 

22.7 

21.0 

18.5 

16.1 

14.1 

12.4 

11.1 

10.0 



4 

18.3 

17.8 

17.4 

17.2 

16.5 

15.2 

13.7 

12.3 

11.0 

10.0 



5 

15.0 

14.6 

14.0 

13.6 

14.0 

13.6 

12.9 

11.9 

10.8 

9.89 


n 

k 

5 = 12 

13 

14 

15 

16 

17 

18 

19 

20 

21 

22 

2 

1 

8.24 

7.63 

7.10 

6.64 

6.23 

5.87 

5.55 

5.26 

5.00 

4.76 

4.54 


2 

7.74 

7.29 

6.87 

6.48 

6.12 

5.79 

5.50 

5.22 

4.97 

4.74 

4.53 


3 

6.51 

6.36 

6.18 

5.97 

5.74 

5.52 

5.29 

5.07 

4.86 

4.66 

4.48 


4 

4.74 

4.88 

4.96 

4.99 

4.97 

4.90 

4.81 

4.70 

4.58 

4.44 

4.31 


5 

3.25 

3.44 

3.59 

3.72 

3.86 

3.96 

4.02 

4.04 

4.04 

4.01 

3.96 

4 

1 

8.33 

7.69 

7.14 

6.67 

6.25 

5.88 

5.56 

5.26 

5.00 




2 

8.31 

7.68 

7.14 

6.66 

6.25 

5.88 

5.56 

5.26 

5.00 




3 

8.22 

7.63 

7.11 

6.65 

6.24 

5.88 

5.55 

5.26 

5.00 




4 

7.94 

7.47 

7.02 

6.60 

6.21 

5.86 

5.54 

5.26 

5.00 




5 

7.34 

7.07 

6.76 

6.44 

6.12 

5.80 

5.52 

5.24 

4.99 




32 


TRAIN BOMBING 


Table 6. Expected number of targets hit at least k times each by each bomber during its lifetime, when s' are dispatched 
on each mission. Risk to bomber at any moment of combat depends linearly on the reciprocal of the number of aircraft 
present; normalization such that probability of surviving a single-aircraft mission is 0.7. Target 1 X 6, 0 = 90°, 
<Ta = 2, <Td = 0-3. 


n 

k 

s' = 1 

2 

3 

4 

5 

6 

7 

8 

9 

10 

11 

12 

4 

1 

1.43 

1.88 

2.18 

2.31 

2.35 

2.35 

2.32 

2.28 

2.24 

2.20 

2.15 

2.11 


2 

0.59 

0.98 

1.39 

1.76 

2.01 

2.15 

2.20 

2.22 

2.20 

2.18 

2.14 

2.10 


3 

0.37 

0.64 

0.93 

1.18 

1.44 

1.71 

1.90 

2.02 

2.08 

2.10 

2.10 

2.08 


4 

0.18 

0.37 

0.60 

0.84 

1.05 

1.24 

1.47 

1.67 

1.82 

1.91 

1.97 

2.00 


5 


0.10 

0.27 

0.48 

0.69 

0.90 

1.09 

1.28 

1.47 

1.63 

1.75 

1.84 

8 

1 

1.97 

2.27 

2.42 

2.45 

2.42 

2.38 

2.34 

2.29 

2.24 

2.20 




2 

1.15 

1.67 

2.11 

2.31 

2.37 

2.36 

2.33 

2.29 

2.24 

2.20 




3 

0.71 

1.14 

1.64 

2.02 

2.22 

2.29 

2.30 

2.28 

2.24 

2.19 




4 

0.51 

0.86 

1.25 

1.64 

1.95 

2.20 

2.22 

2.24 

2.22 

2.19 




5 

0.41 

0.71 

1.01 

1.29 

1.64 

1.90 

2.07 

2.16 

2.18 

2.17 




Having determined the optimum spacing as a 
function of the various parameters, it is desired to 
present this in a form suitable for rapid calculation. 

Method of Analysis. In the analysis, as an aid to 
the discovery of an approximate rule, two quantities 
are read from each graph of the type of Figure 1, 


namely, the spacings, I and I, on either side of the 
optimum spacing I, which correspond to probabilities 
of magnitude 0.99P&. Then any value of the spacing 
in the range from I to I may safely be identified 
with the optimum spacing, 1, without missing the 
maximum probability by more than one per cent. 



Figure 7. Scatter chart showing the empirical invariant on which the Bomh-S pacing Calculator is based. Circles and 
crosses correspond to the least and greatest spacings, respectively, which lead to probabilities of 0.99 Pa. 




RULES FOR BOMB SPACING 


33 


One now tries to synthesize a rule, assisted by this 
latitude thus introduced into the definition of I and 
by qualitative considerations. For example, it is evi- 
dent that the spacing will increase as the target di- 
mensions increase, as the angle of attack decreases. 



Figure 8. Photograph of AMP Boynb-Spacing Calculator. 


as the standard deviation of the aiming-error distri- 
bution increases, and as the number of bombs de- 
creases. 

From this type of argument the following specific 
function has finally arisen for consideration: 

(TgW CSC 
nal ) 

Here the optimum spacing 1 is expressed as an em- 
pirical function of the standard deviations, Ca and 
(jd, of the aiming-error and bomb-dispersion distri- 
butions; of the angle d of attack, measured from the 
long target-axis to the track of the attacking aircraft ; 
of the target width W ; of the number n of bombs in 
train; and of two constants, Ci and Co, which will be 
so chosen as to give the best fit. 

Plotting first I and then I against the expression 
in parentheses in equation (8) for all at-least-one- 
hit data on hand, namely, that corresponding to 
targets of the shapes 1X6 and 1 X 9, 3^ ^ 8, 34 

^ (Td ^ 3, 27° ^ B ^ 90°, 2 ^ n ^ 20 (all linear 



dimensions expressed in terms of target width as 
unit), one obtains a scatter chart like Figure 7. From 
the chart one sees that, over substantial ranges of 
the variables, the two sets of points are nicely sorted 
by the straight line, which represents equation (8). 

Results. The study resulted in the development of 
the Bomb-Spacing Calculator (see Figure 8), which 
mechanizes equation (8). It is a circular slide rule 
designed to provide estimates of the best spacing, 
i.e., that spacing which maximizes the probability of 
at least one hit, in any number of train attacks 
against rectangular targets. It has been calibrated 
with special reference to ship targets. A few of the 
details of the calibration follow. Figures 9 and 10 



Figure 9. Scatter chart of length vs beam for a 
sample of Japanese merchantmen. 

are scatter charts showing the relationship between 
length and beam, and gross tonnage and beam, for 
Japanese merchantmen. From the first, one sees that 
most of these ships have length-to-width ratios which 
fall comfortably within the target limits on which 
Figure 4 is based; the second provides a ready cali- 
bration based on tonnage. 




34 


TRAIN BOMBING 


The component standard deviations, da, of the 
aiming-error distribution are replaced by a new 
aiming-error statistic whose value lies midway be- 
tween that of the so-called circular probable error 
(CEP = 1. 18 (Ta) and that of the mean radial error 


15.000 

10.000 
8,000 

6,000 

4.000 

2.000 


1,000 

800 

600 

400 


200 


100 





o 





o 

«b 









t 

oo^ 

o 

o 




/ 





/ 




■ 1 

s 

o”o° 

oo 

) 




vP 

o 





o 

oo 





° ^ o 

Q Q 




o 

o ooo 

oo 




0 

o 

o 

o 

o c 

c 

O 0 

— Q_ 

0(^0 
o°8 
o° o 
% O 





10 


20 40 

BEAM IN FEET 


60 80 100 


Figure 10. Scatter chart of gross tonnage vs beam 
for a sample of Japanese merchantmen. 


tor being chosen so as to yield dd = 30 ft when 
the altitude is 10,000 ft. 

The bomb-spacing calculator has been manufac- 
tured in sufficient quantities to permit distribution to 
operations analysts and other personnel in the Serv- 
ices who have had use for the device. 

The best spacing for at least k hits, when k > 1, 
has not been mechanized; indeed it has not been 
studied extensively. But from study of a limited set 
of calculations in which the bomb dispersion, dd, was 
substantially less than target width W, a few hints have 
been obtained. For single attacks and for k > n/2, 
the best spacing is zero; forl</c<n/2 the best 
spacing is approximately W jk, but the best spacing is 
always less than that for fc = 1. As the number s of 
attacks is increased, the best spacing increases, until it 
reaches the value for at least one hit; in the cases 
examined this occurred at moderate values of s, like 
s = 10 or 20. 

An interesting side light on the single-attack case 
is provided by the following observation. The prob- 
ability Pk of at least k hits may of course be written as 

n 

Pt = ^ P'i , (9) 

i = k 

where Pi is the probability of exactly ^ hits. It has 
been observed that the spacing which maximizes P'i 
does not depend on the aiming-error standard devi- 
ation, da, except when ^ = 1. Hence, the best 
spacing in single attacks does not depend on the 
aiming-error statistic when k > 1. 

For the basic probabilities, reference may be made 
to the reports listed in sections 3.2 and 3.3 of this 
chapter. Discussions, often brief, of the best spac- 
ing for at least k hits against a single target are 
contained in AMP working papers^^’^^’^^ and an 
AMP report.^ 


3 5EMPIRICAL RULES FOR DETERMINING 
THE PROBABILITIES OF HITTING 
RECTANGULAR TARGETS 


{MRE = 1.25 (7 a). Thus the aiming-error statistic 
will not be biased by more than 3 per cent if the slide 
rule is entered with CEP or with MRE. 

The bomb-dispersion statistic is contained im- 
plicitly: It is assumed that dd is proportional to the 
square root of the altitude (available data support 
this weakly, but then the slide rule depends only 
weakly on the assumption), the proportionality fac- 


The bomb-spacing calculator described in Section 
3.4 under Results shows the best way to space a 
train of bombs, from the viewpoint of at least one 
hit, but it does not indicate how good this best way 
is, measured in probabilities. The present study con- 
cerns an attempt to provide such estimates. How- 
ever, it was undertaken at a time when the impor- 
tance of this phase of the work was judged to be 


RULES FOR DETERMINING PROBABILITIES OF HITTING 


35 


secondary; consequently, the specifications were not 
made very stringent. 

Purpose of the Study. The purpose of the study is to 
try to provide a simple method, or device, for esti- 
mating the probability Pks of at least k hits when s 
attacks are made on a single target, or, conversely, of 
the number s of attacks needed in order to have Pks 
attain a specified value. 

From the viewpoint of subsequent mechanization 
it is highly desirable that an empirical formula be de- 
veloped which can be written as a product of factors, 
each of which depends on only one of the parameters 
s, k, n, etc. 

Method of Analysis. Two analyses of this type have 
been made, one based on 

<7iW = g2{Pks)gs(k)g4(n)g3i(Ta) , (10) 



Figure 11. Design of multiple-attack multiple-hit 
slide rule. T = 1 X 6, 0 = 90°, aj, = 0.3. 


for a given target and Ud, with emphasis on values 
of Pks ^ 0 .75; and one based on 


— [=fr 


( 11 ) 


for /c = 1, targets 1X6 and 1X9, and for Ud ^ 2. 

Results. As a result of the study of spacing, the 
functions gi{i = 1, • • • , 5) in equation (10) have 
been determined empirically for the 1X6 target, 
(Td = 0.3; and a mockup of a multiple-attack mul- 


tiple-hit slide rule, based on that equation, has been 
constructed. This is exhibited in Figure 11. When 
used to estimate s the error rarely exceeds 20 per cent. 
To use the slide rule, the selected value of k on the 
k scale is matched against the selected value of Pk 



Figure 12. Photograph of AMP M ultiple- Attack 
Bombing Calculator. 


on the Pk scale. The appropriate value of n is then 
matched against the value of Ua. The arrow will then 
indicate the value of s, the number of attacks to make. 

The constants Ci and C 2 in equation (11) have 
been determined from data covering a wider range 
of values of Pks than was used in equation (10), 
and from all the data at hand regarding ship-like 
rectangular targets. The equation has been mecha- 
nized in a slide rule known as the Multiple- Attack 
Bombing Calculator, shown in Figure 12. When used 
to estimate s, errors as great as 50 per cent have been 
observed. 

This slide rule was deliberately restricted to the 
problem of at least one hit {k = 1) in order that it 
could be used freely, with little danger of misuse, as a 
comparison instrument to the bomb-spacing calcu- 
lator (see Section 3.4 under Results); the latter 
gives the spacing which maximizes the probability 
of at least one hit (A; = 1). 

The multiple-attack bombing calculator has been 
manufactured in quantities sufficient to permit dis- 


C 



36 


TRAIN BOMBING 


tribution to operations analysts and other personnel 
in the Services who have need for the instrument. 

Documents by AMP’s Bombing Research Group 
[BRG] cover the study discussed in this section in 
greater detail. 

36 MISCELLANEOUS INVESTIGATIONS 
OF PROBABILITIES OF HITTING 
RECTANGULAR TARGETS 

There are discussed here a number of auxiliary 
questions which arose in the course of the train- 
bombing investigations, questions which still seem 
to have value. No mention is made of those questions 
of transitory interest on which, in an extended in- 
vestigation, time is inevitably dissipated. 

Purpose of the Study. The object of the study was 
to answer the four auxiliary questions listed here: 

1. Efficiency. How do the probabilities of hitting 
in train bombing compare with those for certain other 

Table 7. The probabilities of at least one hit when 
n bombs are released 1, 2, 4, or n per bombing run. The 
train releases are made at optimum spacing. Target 


1 X 6, 0 

= 90°, (Ta 

= S, ad = 0.5. 



Total bombs 

Number of bombs released per bombing run 

n 

1 

2 

4 

n 

1 

0.01 



0.01 

2 

0.03 

0.03 


0.03 

4 

0.06 

0.06 

0.06 

0.06 

8 

0.11 

0.11 

0.11 

0.10 

12 

0.16 

0.16 

0.16 

0.14 

16 

0.21 

0.21 

0.21 

0.17 

20 

0.25 

0.25 

0.25 

0.20 


methods in which the same number of bombs and /or 
-aircraft are employed? 

2. Offset. How serious is the effect of aiming the 
first bomb of a train at target center instead of aim- 
ing the center of the train? 

3. Errors. How seriously will mis-estimates of the 
standard deviation era affect the planning and ex- 
ecution of a mission? 

4. Combat data. How may these standard devia- 
tions (Ta be estimated from combat data? 

Method of Analysis. The methods of analysis are 
either evident from the discussion of the preced- 
ing studies, or implicit in the results which will be 
exhibited. 


Results. The results of the study are listed as 
answers to the four questions presented in Purpose 
of the Study. 

1. Efficiency. We shall compare the probabilities of 
at least one hit when the same total of bombs is 
dropped in various ways (but under the same con- 

Table 8. Comparison: 12-aircraft formation vs s inde- 
pendent attacks with trains of 12 bombs from point of 
view of probability of at least one hit. Modeled on 
high-altitude combat data. 6 = 90°. 


(Ta 

<Td 

Target 

12-aircraft 

formation 

s = 1 

s = 5 

s = 6 

s=7 s=8 

8 

0.86 

1 X 9 

0.62 

0.18 

0.62 





1 X6 

0.58 

0.12 

0.49 

0.55 

0.61 

16 

1.7 

1 X9 

0.26 

0.05 


0.26 

0.30 



1 X6 

0.19 

0.03 



0.18 0.20 


ditions regarding aiming errors, etc.), namely, single 
releases, trains of two bombs each, trains of four, 
and a single train containing all the bombs. 

The calculations show that the tendency is for the 
probability to decrease when there are fewer aiming 
operations. However, when n is small there is usually 
very little difference between the probability of at 
least one hit with n single releases and that with one 
train of n bombs; for large values of n the difference 
more often becomes sizeable. Also, there are cases in 

Table 9. Probability of at least k hits with s trains of 
8 bombs when (a) train center and (b) the first bomb 
are aimed at target center. Target 1 X 6, 0 = 90°, 

<Ta = 2, ad = 0.3. 


k 

I 

s = 1 

(a) (b) 

s = 2 

(a) (b) 

s = 4 

(a) (b) 

s = 8 

(a) (b) 

1 

0.4 

0.58 0.48 

0.82 0.73 

0.97 0.93 

1.00 1.00 


0.8 

0.71 0.47 

0.91 0.72 

0.99 0.92 

1.00 0.99 

2 

0.4 

0.41 0.34 

0.68 0.58 

0.92 0.85 

1.00 0.98 


0.8 

0.24 0.14 

0.64 0.38 

0.94 0.73 

1.00 0.96 

3 

0.4 

0.19 0.15 

0.47 0.37 

0.82 0.71 

0.99 0.96 


0.8 

0.01 0.00 

0.29 0.12 

0.81 0.47 

1.00 0.89 

4 

0.4 

0.04 0.03 

0.27 0.19 

0.69 0.56 

0.97 0.91 


0.8 

0.00 0.00 

0.07 0.02 

0.58 0.24 

0.98 0.75 

5 

0.4 

0.00 0.00 

0.14 0.09 

0.54 0.40 

0.94 0.85 


0.8 

0.00 0.00 

0.00 0.00 

0.32 0.09 

0.93 0.57 




MISCELLANEOUS INVESTIGATIONS 


37 


which a train is definitely better than the same num- 
ber of bombs released singly, a point which has not 
been generally appreciated. Table 7 is a typical ex- 
ample selected from a set of similar tables. 

Comparison may also be made between inde- 
pendently aimed trains and trains dropped on the 
leader’s signal. Table 8 shows the number of inde- 
pendently aimed trains required to match a combat 
box of 12 aircraft dropping on the leader. This exam- 
ple is modeled on the Eighth Air Force’s experience 
in the European Theater of Operations. 

2. Offset. The effect on Pk, the probability of at 
least k hits, of aiming the first bomb of a train in- 
stead of aiming the train center is always deleterious 
and at times serious. However, no simple general 


Table 10. Number s of attacks needed to insure that 
maximum probability Pks of at least k hits will exceed 
0.90. Target 1x6, d = 0°, ad = 0.25. 


k 

n = 2 

aa = 1 

n = 4 

71 = S 

n = 2 

tTa = 2 

n = 4 

n = 8 

1 

4 

3 

3 

10 

8 

7 

2 

6 

4 

4 

14 

10 

8 

3 

8 

5 

4 

18 

12 

9 

4 

10 

6 

4 

23 

14 

10 

5 

11 

7 

5 

>25 

17 

11 


rule has been discovered for isolating the serious 
cases, in which the biased aiming operation may re- 
duce Pk by one-half or more. Table 9 shows a sample 
comparison of the two methods of aiming. 

It should be noted that these comparisons are 
based on the assumption that the aiming-error dis- 
tribution, relative to its mean, is the same with each 
method of aiming. In view of the limitations imposed 
by our bombsights, the aiming-error statistic a a is 
probably larger when train center is aimed; hence, 
calculations such as those displayed in Table 9 tend 
at present to overestimate the importance of the 
effect. 

3. Errors. Mis-estimates of the distribution-func- 
tion parameters, a a and ad, can be expected to pro- 
duce two effects: (1) the mission planning will be 
upset in that an improper force will be assigned to 
the target; (2) the force assigned will not, because of 
mis-information, do the best job of which it is capable. 

The results of the study indicate that the first 
effect is generally the more serious; force require- 
ments depend with particular sensitivity on the aim- 
ing-error distribution. This is illustrated in Table 10, 


which indicates the number s of attacks needed to 
insure a probability of success equal to 0.90, when 
the standard error of aim, a a, is 100 ft and 200 ft, 
respectively. This table is based on the best spacing 
in each case. An overestimate of (Ta may be less 
serious (and an underestimate more serious) than 


Table 11. Efficiency of attack when aa is mis-esti- 
mated, judged by ratio of probabilities of at least k hits. 
Target 1x6, d = 0°, ad = 0.25. 


True 


200 



100 


<ra 









Assumed 

100 



200 


k \ 

1 

2 

4 

8 

1 

2 

4 

8 

1 

1.00 

0.98 

0.97 

0.95 

1.00 

1.00 

0.99 

0.99 

2 

0.75 

1.00 

1.00 

0.97 

0.71 

1.00 

1.00 

1.00 

3 

0.59 

0.71 

0.97 

1.00 

0.63 

0.68 

1.00 

1.00 

4 

0.50 

0.67 

0.94 

1.00 

0.58 

0.68 

0.80 

1.00 

5 

0.12 

0.58 

0.76 

1.00 

0.56 

0.58 

0.68 

1.00 


suggested by the table, for the mis-estimate will prob- 
ably be accompanied by the use of other-than- 
optimum spacing, which will diminish (or enhance) 
the apparent difference. 

The effect of mis-estimate on the execution of the 
mission is less pronounced, for here one is not con- 
cerned with the difference between what can be 
accomplished with one value of aa (or ad) and 
another value of aa (or ad), but only with the differ- 
ence between what can be accomplished when it is 
and is not recognized that aa (or ad) has a certain 


Table 12. Efficiency of attack when ad is mis- 
estimated, judged by ratio of probabilities of at least 
k hits. Target 1 X 6, 0 = 90°, aa = 2. 


True 

ad 

Assumed 

0.3 

0 


0 

0.3 

k \ 

1 

2 

4 

1 

2 

4 

1 

1.00 

0.96 

0.99 

1.00 

0.97 

0.94 

2 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

3 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

4 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 

5 

1.00 

1.00 

1.00 

1.00 

1.00 

1.00 


38 


TRAIN BOMBING 



Figure 13. Synthetic bomb plot illustrating technique for measuring standard deviation of aiming errors from post- 
raid photograph. 




APPLICATIONS OF RECTANGULAR TARGET THEORY 


39 


value. Mis-information regarding da (or era) will 
cause one to use an incorrect spacing, but as observed 
earlier, the curves Pk versus I are generally so flat- 
topped that substantial departures from the optimum 
I do not necessarily imply substantial changes in Pk. 
Illustrative cases are exhibited in Tables 11 and 12. 

4. Combat data. A method of estimating combat aim- 
ing errors, applicable if the bomb plot is not biased, 
is given here. If each of s aircraft independently aims 
a train of n bombs and if a post-raid photograph 
shows some of the craters, say N, an estimate may 
be made of da, the one-dimensional standard devi- 
ation of the aiming-error distribution, provided it 
can be assumed that the aiming-error distribution 
is circularly symmetric and that the parameter da 
specifying the bomb-dispersion distribution is known. 
In fact, da may be estimated from 

= ( 12 ) 


where the symbols r and R are defined as follows: 
Choose an origin 0 near the center of the observed 
bomb fall; measure the distance, Ri^ from 0 to Ai, 
the intended point of impact for the fth bomb; 
measure also the distance from 0 to «/, the actual 
point of impact of the fth observed crater. Then 


and 


ns 



(13) 


(14) 


{ns is the number of bombs dropped and N is the 
number of craters counted). 

The situation is illustrated in Figure 13 which 
shows N = 100 craters selected at random from 
ns = 4 X 100 bombs dropped by two waves of 50 
aircraft, each aircraft carrying four bombs. The aim- 
ing points for the bombs in train are designated by 
Ai, A 2 , A3, A 4 for one wave, and by A 5, A e, A 7, A g for 
the other. The plot was synthesized from a certain 
train-bombing experiment performed at Eglin Field. 
In this case equation (12) gave da = 217 ft compared 
to the value 224 ft when all the data were used in 
the most efficient manner. 

The details of the material here are contained in a 
number of AMP papers. 


37 APPLICATIONS OF RECTANGULAR 
TARGET THEORY 

The present section comprises reviews of several 
specific applications of train-bombing theory, appli- 
cations both to high- and to low-altitude bombing. 

The principal distinction between high- and low- 
altitude bombing, from the point of view of theory, 
is that in one case the target may be regarded as a 
planar region and in the other it usually may not, 
for at low altitudes the bomb trajectories at impact 
depart from the vertical to such an extent that the 
three-dimensional character of the target often can- 
not be neglected. 

Another distinction, which frequently occurs when 
the Norden and Sperry synchronous bombsights are 
used, concerns the aiming-error distribution. This is 
almost a circular Gaussian at high altitudes, whereas 
at low altitudes the range component greatly exceeds 
the deflection component. 

Purpose of the Study. The purpose of this investi- 
gation was to study the effect of rectangular target 
theory on four types of targets. 

1. Bridges. To determine the probability Pi of at 
least one hit and the best spacing I in high-altitude 
attacks on bridge or viaduct-like targets. 

2. Ships: Hitting. To determine the spacing best 
suited to produce at least one hit in low-altitude 
radar-sighting attacks on shipping targets. 

3. Ships: Sinking. To determine the probability 
of sinking, as opposed to hitting, in attacks on ship- 
ping targets. 

4. Minefields. To determine the number of bombers 
(light, medium, or heavy) which must attack a mine- 
field in order that the probability of clearing a pro- 
portion P of a path shall be 0.5 or 0.9. 

Method of Analysis. The analysis was done sepa- 
rately for each of the four categories. 

1. Bridges. The basic tables for the probabilities of 
at least one hit are extended so as to provide infor- 
mation on 1 X 6, 1 X 9, 1 X 13, 1 X 20, 1 X 30, 
1 X 00 targets. Nomograms are constructed from 
which the probability Pi and the best spacing 1 
can be read for any high-altitude attack, on the as- 
sumption that da = 34 mils {CEP = 40 mils) and 
(Td = 4 mils. 

2. Ships: Hitting. The best spacings I to produce 
at least one hit are explored in the basic train- 
bombing tables, and then suitably averaged. Account 
is taken of the altitudes, aircraft speeds, aiming- 
error distributions, bomb ricochet, and ship types 


40 


TRAIN BOMBING 



Figure 14. Nomogram for estimating best spacing t of n = 8 bombs in train, and probability Pi of at least one hit 
for single attack on bridge. The numbers along the curves indicate the ratio of bomb s])acing to altitude. 


common to APQ-5 radar bombsight attacks against 
Japanese shipping, as well as of the fact that the 
angle of attack 6 is usually not known. 

3. Ships: Sinking. If a ship is hit by a bomb, there 
is a probability p that it will sink. If it does not sink 
as a result of the first hit, empirical data suggest 
that the effect of a second hit is approximately inde- 
pendent of the first, i.e., that again the probability 
is p; and hence, in general, that the probability of 
sinking, given i hits, may be approximated by 

1 - (1 - pY. (15) 

If the probability of obtaining exactly i hits in s 
attacks is P'is, then the probability of sinking a 
ship is 

1 - (1 - p)* ■ (16) 

4. Minefields. The problem of minefield clearance 
is solved using a model experiment: 50 synthetic- 
train stencils (incorporating bomb dispersion) are 


prepared. A stencil, selected at random, is placed on 
a map of the minefield in such a manner that its cen- 
ter will fall on a mark indicating an aiming error 
drawn from a known Gaussian distribution. The 
range and deflection components of aiming error, 
measured by (Tar and (Tad, are sometimes taken to 
be unequal. The process of placing stencils on the 
minefield continues with periodic inspections to 
determine the proportion F of clearance achieved 
along the best path. The radius of clearance, depend- 
ing both on bombs and mine, is involved in this cal- 
culation. The complete experiment is replicated 30 
times for each set of conditions. Further details of the 
problem of minefield clearance are given in Chapter 7. 

Results. Most of the results of the study are given 
in tables and charts. The results are itemized for 
each of the four categories. 

1. Bridges. The principal results for attacks on 
bridge-like targets are contained in nomograms, of 
which Figure 14 is an example. This nomogram ap- 
plies to the case of n = 8 bombs in train and, like 




SCATTER-BOMBING THEORY 


41 


all the nomograms, to the case da = 34 mils, 0-^=4 
mils. To use it one places the index arrow of the de- 
tachable altitude scale against the horizontal target- 
width scale. Then, at the appropriate altitude mark, 
one erects a perpendicular which intersects the curves 



0 20 40 60 80 100 

F 


Figure 15. Percentage clearance F of best path 
through minefield vs number s of attacks by medium 
bombers needed to give 50 per cent confidence in result. 
Target 6 X <», 0 = 90°, <Ta = Q, aa = 0.3, I = 0.3, 
n = 6, width of path = 0.3. 

corresponding to various target ratios, 1 X 6, 1 X 9, 
etc. Opposite the appropriate intersection, on the 
vertical scale, one reads Pi, the probability of at 
least one hit with a single train of 8 bombs. The 
numbers along the curves are the ratios of bomb 
spacing to altitude. An auxiliary table^^ gives Pi«, 
the probability of at least 1 hit with s trains, given Pi. 

2. Ships: Hitting. The results for low-altitude at- 
tacks ( < 800 ft) using the APQ-5 radar bombsight 


are contained in a simple rule-of-thumb for bomb 
spacing, stated in Table 13. The probability of hit- 
ting when this rule is used is something like 25 per 
cent greater than when a spacing equal to target 
width is used. 


Table 13. Rule-of-thumb for bomb spacing / in APQ-5 
attacks on shipping targets. 


Bombing from a 

Using as spacing the 
ship’s beam multiplied by 

PBY 

2.1 

PBM 

2.3 

PB4Y or PBJ 

2.6 

PV2 

3.1 


3. Ships: Sinking. The principal conclusion re- 
garding ship-sinking probabilities is that the best 
spacing of bombs in train is somewhat smaller than 
I required to maximize the probability Pi of at least 
one hit. While no general rule has been discovered, 
it appears that the best spacing usually lies in the 
range ^ / to 7. In view of the tendency for P versus I 
curves to be relatively flat-topped, the spacing 1 
which maximizes the probability of at least one hit 
often nearly maximizes the probability of sinking. 

4. Minefields. The principal results are given in 
graphs like Figure 15. Here, for given conditions re- 
garding the aiming-error and bomb-dispersion dis- 
tributions, the number of bombs in train, aircraft 
type, width of path and of minefield, there is a plot 
of proportion F of clearance along best path versus 
number of aircraft, with radius of clearance as the 
family parameter. 

The study includes similar results for pattern 
bombing. 

Full discussions of these studies are contained in 
several AMP documents.^^’^^’^^’^^’^^ 

3 8 SCATTER-BOMBING THEORY 

The process of releasing a number of bombs simul- 
taneously from approximately the same position in 
space is sometimes called scatter bombing. This is, 
in a sense, a transition stage between train bombing 
and pattern bombing, for it may be regarded as train 
bombing in which the spacing I is zero, or it may 
be regarded as pattern bombing in which the dimen- 
sions of the formation of aircraft (the usual imple- 
ment in pattern bombing) are negligible compared 
to the pattern dimensions. 


42 


TRAIN BOMBING 



I 



0 12 3 4 

I 


Figure 16. Probability Pi of at least one hit (graph A) and expected number E of hits (graph B) vs spacing I of 
bombs in train, for various values of 6, the angle of attack. Target 1 X 6, o-a = 8, <rd = 0, n = 8. 


The theory presented below usually presupposes a 
quite limited number of bombs and always pre- 
supposes a circular-Gaussian distribution, character- 
ized by ddi about the center of each cluster. Since 
these conditions are met more frequently in the case 



0 12 3 4 5 

CTd 

Figure 17. Probability Pi of at least one hit and 
expected number E/n of hits per bombs vs standard de- 
viation (Td of bomb dispersion, for values of n, the num- 
ber of bombs in salvo. E/n given by broken line. 
Target 1 X 6, o-a = 4. 

of single aircraft, we have preferred to discuss scat- 
ter bombing in the present chapter on train bombing, 
rather than in Chapter 4, which is devoted to pattern 
bombing. 

Since scatter bombing may be viewed as the limit- 
ing case of train bombing, in which the spacing 7=0, 


and since it has been demonstrated that the optimum 
spacing I which maximizes the probability P* of 
at least k hits is not in general zero, the question 
may be asked : Why is there interest in scatter bomb- 
ing? The answer is twofold: (1) It is not always pos- 
sible to capitalize fully on the potentialities of train 
bombing, as when small bombs are released from a 
cluster. (2) It is not always desirable to do so, for 
scatter bombing maximizes the expected number of 
hits, or the long-term average number, say E. 

The last point may be demonstrated by reference 
to Figure 16 where the graph A illustrates the famil- 
iar dependence of Pi on I and B (the discontinuities 
in the derivatives are characteristic of the special 
case where bomb dispersion da is zero), and graph B 
shows the dependence of the expected number E of 
hits on I and E does not depend on the angle of 
attack and it attains its greatest value when 7=0. 

Purpose of the Study. The purpose of the stiidy is 
to provide basic values of the probability Pi of at 
least one hit and of the expected number E of hits 
for rectangular and circular targets under conditions 
of scatter bombing. 

Method of Analysis. The method of analysis is again 
that of formal probabilities. In fact, for the rectangu- 
lar targets, equations (1) through (3) suffice if the 
spacing 7 is set equal to zero. 

For circular targets the formulas may be written 




2ir 

e 


(— l/2<rh(p* + a;* — 2px cos d) 


pdddp. (18) 


rnTiTFTrrnTTi^^n ^ 


SCATTER-BOMBING THEORY 


43 


Here, R is the radius of the target and n the number 
of bombs, and da and dd are the standard deviations 
which characterize the circular-Gaussian distribu- 
tions of the aiming errors and of bomb dispersion, 
respectively. 

Extensive tables of q have been prepared in the 
course of the work and a number of approximate for- 
mulas have been developed. 

Results. The principal results of the study are pre- 
sented in the form of graphs. Figure 17 is a typical 



Figure 18. Scatter-bombing attack: Probability Pi 
of at least one hit vs ratio, crd/o-a, of standard deviations 
of bomb dispersion and aiming errors, for various values 
of R/(Ta', n = 8. 

example of the rectangular-target results, where the 
values of Pi and of E are shown by solid and broken 
lines, respectively. It should be noted that Efn, 
rather than E, is plotted; the reasons for this choice 
being that E/n is invariant with respect to n and that 
this quantity may be plotted on a zero-to-one scale, 
as if it were a probability. 

It will be observed that, in a certain sense, the 
bomb dispersion, da, plays a role in scatter bombing 
analogous to the spacing I in train bombing. The 
spacing related to dd is, superficially, more of a sta- 
tistical affair than is 7, which is usually — and errone- 
ously — thought of as a strictly geometrical effect. As 
in the case already discussed for 7, there is an opti- 
mum value of dd, say dd. 



0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 

‘^dAo 


Figure 19. Scatter-bombing attack: Probability Pi 
of at least one hit vs ratio, aaha, of standard deviations 
of bomb dispersion and aiming errors, for various values 
of n, the number of bombs in salvo; R = a-a. 



Figure 20. Scatter-bombing attack: Probability Pi 
of at least one hit vs ratio, R/aa, of target radius to 
standard deviations of aiming errors, for various values 
of n, the number of bombs in salvo; ad = aa. 


C' 


44 


TRAIN BOMBING 


Calculations for the rectangular targets have been 
made for the following conditions: targets 1X1, 
1 X 3, 1 X 6, 1 X 9; (7a = 1,248 (and 16 for 1 X 9 
target) ; n = 2, 4, 8, 12, 16, 20. 

The results for the circular targets are presented in 



0 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 

Figure 21. Scatter-bombing attack: Expected num- 
ber Ejn of hits per bomb vs ratio, (Tdl<Ta, of standard 
deviations of bomb dispersion and aiming errors, for 
various values of R/cra. 

several sets of charts. Figures 18, 19, and 20 being 
typical examples of each. All sets are based on the 
same data, the various sets having been compiled as 
a convenience to the user, who may find that one or 
another set is best suited to his needs. 

Figure 18 shows Pi versus cTdlcTa, with R/aa as a 
family parameter; Figure 19 uses n as the family 
parameter. Figure 20 shows Pi versus R/aa, with n 
as the family parameter. The ranges of the variables 
covered are 1 to 20 for n, to 3 for P, 0 to 2 for ad 
(the last two are in units of a a). 

Figure 21 shows Ejn versus adjaa, with Rjaa as 
the family parameter. This chart, thanks to the 
invariance of Ejn with respect to n, displays all the 
results, concerning expected number of hits, calcu- 
lated for circular targets. 

Figure 22 is a special chart for the optimum dis- 
persion, ad, optimum with reference to the maximum 
probability Pi of at least one hit (the optimum value 
of (7d is 0 with reference to E). Here is shown a dj a a 
versus P/(7a, with n as the family parameter. 


The theory and results outlined above are reported 
more fully in AMP publications. 

39 APPLICATIONS OF SCATTER-BOMBING 
THEORY 

Very few applications of the scatter-bombing the- 
ory discussed above have appeared in bombing opera- 
tions during the course of World War II; for although 
bombs have frequently been dropped in clusters, 
e.g., the small incendiaries, these have almost invari- 
ably been employed as trains of clusters, or as pat- 
terns of clusters. 

The theory has had applications, however, in the 
field of aerial gunnery and in the field of naval 
gunnery. As a matter of fact, the computations de- 
scribed in Section 3.8 were undertaken at the request 
of the Bureau of Ordnance. 

The applications to bombing described below, 
namely air-to-air and guided-missile bombing, are 
somewhat special. In fact, they derive only minor 
assistance from the calculations reported above. 

Purposes of the Studies. The purposes of the studies 
are: 

1. Air-to-air. To obtain a rough estimate of Pi 
where a single B-29, or a four-aircraft diamond of 



Figure 22, Scatter-bombing attack: Ratio, o-d/o-a, 
of standard deviations of optimum bomb dispersion to 
aiming errors vs R/cra, for various values of n, the num- 
ber of bombs in salvo. 

B-29’s, is attacked with a 50-bomb cluster, per- 
cussion fuzed. 

2. Guided missiles. To obtain a rough estimate of 
Pi when a long narrow target is attacked with sev- 
eral AZON (a missile whose position in deflection 
may be modified by remote control), released simul- 
taneously from a single aircraft. 




APPLICATIONS OF SCATTER-BOMBING THEORY 


45 


Method of Analysis. The analysis was done in terms 
of air-to-air and guided-missile bombing. 

1. Air-to-air. The target area in the case of a B-29, 
or a diamond formation of B-29’s, is extremely com- 
plicated compared to the simple rectangles and circles 
considered above. Therefore, no direct use may be 
made of the probabilities so far calculated. However, 
the latter can be made to yield a hint regarding the 
magnitude of the optimum dispersion, ad. 

Using an estimated value of ad and a table of 
random numbers, a pattern of 50 bombs is con- 
structed. For the aiming-error distribution of inter- 
est (a circular Gaussian with components), a a, a 
sample of n aiming errors is constructed, again using 
a random-number table. The sample bomb pattern 
is now centered in turn at each of the aiming-error 
points, which are marked on a map of the target, 
and at each position of the pattern one observes 



but he cannot reduce the inherent scatter in deflec- 
tion, measured by ad. Assume that he chooses a 
bomb at random and guides it to the line target. The 
aiming errors associated with this operation are 
measured by aa. It is an experimentally determined 
fact that (7a << (7rf. Under these conditions the prob- 
ability of at least one hit is approximately 

Pi = 1 - [1 - p((7a)] [1 - p{^2ad)y -^ , (19) 

where p{a) is the probability of hitting with a single 
AZON when the standard error of aim is a. The 
factor, V 2, arises because we are now interested in 
the distribution of the cluster about an arbitrary 
member of the cluster, rather than in the distribution 
about its natural center. 

Results. The principal results are given in tables 
and charts. 

1. Air-to-air. The results of the study are summed 
up in a single graph. Figure 23, where the probability 
Pi of at least one hit is plotte*d against the standard 
deviation aa of the aiming errors. 

It appears that, with even a moderately good solu- 
tion to the aiming problem, this tactic constitutes 
a real threat to bombardment aircraft. However, the 
probability for hitting can probably be diminished 
greatly if the target has any freedom for maneuver. 

2. Guided missiles. Table 14 shows the result of 
three calculations based on equation (19). Experi- 


0 200 400 

Figure 23. Probability Pi of at least one hit vs 
standard deviation aa of aiming errors, for scatter- 
bombing attack on one B-29 and on a four-ship diamond 
of B-29’s. Attack made with 50 percussion-fuzed bombs 
having approximately optimum dispersion. Diamond 
formation assumed to be 200 ft wide between centers; 
aircraft flying nose-to-tail. 

whether or not there is a hit. The success ratio in a 
series of trials is an estimate of Pi, i.e., Hjn-^ Pi, 
where H is the number of trials which yielded hits. 

2. Guided missiles. The estimation of Pi for a salvo 
of AZON is relatively easy under the circumstances 
postulated here. Several AZON, say n, are released 
simultaneously from an aircraft flying parallel to a 
long target and then controlled by a bombardier 
using a single control box; all AZON receive the 
same signals. Thus, the operator may guide the 
centroid of the cluster to a desired position, or he 
may attend to any particular member of the cluster. 


Table 14. Probability Pi of at least one hit with salvo 
of six AZON. For one AZON, P\ = 0.26; for six indepen- 
dent attacks, Pi6 = 0-84. Target = 0.2 X <ra = 0.3. 


^ 2 ad 

Pi 

100 

0.51 

200 

0.40 

400 

0.33 


mental data suggest that V2(7d is of the order of 
300-400 ft. 

Since the cost of bombs is a small part (a few per 
cent) of the overall cost of a bombing mission, it is 
definitely worth while to carry and use a full load 
of AZON even on single-bombing-run missions. 

The work on air-to-air bombing is more fully de- 
scribed here than in the SRG-P working paper,^^ 
which is the only written record. The AZON work 
is discussed in an AMP report.^^ 


Chapter 4 

PATTERN BOMBING 


41 INTRODUCTION 

T he principal bombing tactic used by the United 
States medium heavy and very heavy bombers 
in World War II employed the almost simultaneous 
release of all the bombs carried by a formation of 
aircraft, thus giving rise to a pattern of bombs 
affected, as a unit, by an aiming error. This pattern 
is the fundamental unit in terms of which such bomb- 
ing is discussed. 

The Army Air Forces developed this tactic to a 
surprisingly high state of efficiency in visual oper- 
ations, considering the inherently difficult coordina- 
tion requirements and the instrumental limitations. 
The ultimate in this kind of bombing would be repre- 
sented by the ability to lay a pattern of the size 
wanted at the place wanted. The first requires pre- 
cision flying by all pilots and prompt release by all 
bombardiers. (The radio-release operating from the 
lead aircraft was used in the last year of the war, 
but rarely even then.) The second requires a bomb- 
sight capable of accurately aiming a pattern, rather 
than a single bomb, and of aiming it at a target 
offset, if necessary, from a good aiming point. The 
radar operations, of course, have even greater need 
for such a bombsight. 

Early in the war the bombing studies of AMP 
were, as the result of stimulus from the Army Air 
Forces, largely devoted to train bombing. As the 
war progressed and the Army Air Forces developed 
the art of pattern bombing, AMP, after some lag 
due to insufficient information from the Services as 
well as to a desire to finish the train-bombing work 
at hand, shifted the emphasis in its work to pattern 
bombing. By the time the war ended AMP’s efforts, 
being devoted to the various kinds of bombing, bore 
a reasonable relationship to the relative frequencies 
of these forms in combat. 

4 2 average proportion of hits with 

UNIFORM PATTERNS 

The most common criterion of success in pattern 
bombing is the expected proportion, or long-term 
average proportion, of hits. Reference to scatter- 
bombing theory shows that this criterion is maxim- 
ized by making the pattern as small as possible. It 


is evident that one would become dissatisfied with 
this method of measuring success if the patterns were 
made very small, for it would then attach a premium 
to over-bombing. Probably a better criterion would 
be the proportion of target elements hit, but to cap- 
italize on this concept would require tailoring the 
pattern to the target, a process which would have 
been very difficult with Army Air Force equipment 
of World War II vintage. 

Purposes of the Study. The purposes of the study 
were to calculate the expected proportion E(H) of 
hits and the standard deviation (Th oi the propor- 
tion for various values of the pattern dimensions and 
mean radial error MRE (or circular probable error 
CEP or standard deviation of aiming error, cja). 

Method of Analysis. The mathematical model is 
that of a rectangular target. It X Wt, and a uniform 
rectangular bomb pattern. Ip X wp, with the sides 
of the latter parallel to those of the former. The 
center of the pattern is subject to a Gaussian dis- 
tribution of errors with its mean at the target center. 

The expected proportion of hits is 

[P(F0 - P(TF2)] 
( 1 ) 

and the variance is 

ah = Em - E\H) , ( 2 ) 

where 

«"*> - (s)’ [«“ - - 1 ««] 

- SCTTi.) - . (3) 

The parameters so far undefined are identified 
below. 


M = ka a 


Li,L2 


Ip Ip 
Ml 


F],TF2 


Wp i Wp 

Ml 


(4) 


I = min (Ip, Ip) ; w = min {wp,wp) . 


46 


PROBABILITY OF PROPORTION OF HITS 


47 


The functions P(x) and S{x) are defined as follows 
when A: = 1 (i.e., M = (To) > 



Results. The functions, P{x) and S(x), in terms of 
which E{H) and an are expressed, have been tabu- 
lated by AMP for use with a a {k = 1), MRE 
{k = 1.2533), and CEP (k = 1.1772). Two or three 
Army research groups have independently tabulated 
either P{x) or E{H), for the one-dimensional case. 
Graphs and tables are available for G{x) and G\x). 

A nomogram for E(H) is shown in Figure 1. While 
ostensibly designed for use exclusively for square 
patterns and square targets, it may be applied to 
rectangular patterns and rectangular targets by en- 
tering first with the lengths, say, then with the 
widths. The desired answer is the geometric mean 
of the two values of E(H) so found, i.e., the square 
root of the product of the two values of E(H). 
Errors as great as about 12 per cent have been ob- 
served in the nomogram, but generally the error is 
about 5 per cent. 

Contour diagrams for E{H), of the type shown in 
Figure 2, are available for square targets, rectangular 
patterns, and equal and unequal components of aim- 
ing error, (Tar and dad^ The latter charts are of value 
in assessing pattern bombing with the controlled 
missile, AZON. 

The graph in Figure 3 provides estimates of <Jh, 
but only for square patterns and square targets. 

The tables and graphs referred to and more de- 
tailed presentation of the theory may be found in 
several AMP documents. 

4 3 probability distribution of the 

PROPORTION OF HITS WITH UNIFORM 
PATTERNS 

The expected proportion, E(H), of hits and the 
standard deviation, cth, calculated in the preceding 
study, do not tell the whole story. For example, one 
may wish to know the probability that at least a 



EXPECTED 
PROPORTION OF 
HITS 
-- 0.95 
0.9 

-- 0.8 

0.7 

0.6 

-- 0.5 

0.4 

0.3 


r - 4.0 


- 3.0 


- 2.0 



0.2 


0.1 

0.08 

0.06 

0.04 


±L 

MRE 



0.02 


— 1.0 
1 0.9 


— 0.8 


0.01 - 0.7 

0.008 

0.006 _ 0.6 

0.004 


0.002 


- 0.5 


0.001 


_ 0.4 


- 0.3 

- 0.25 

Figure 1. Nomogram for estimating the expected pro- 
portion of hits when square patterns (side = Ip /MRE) 
are released against square targets (side = It/MRE). 


given proportion of hits, H, will occur, when H is 
assigned at pleasure. It was with this thought in 
mind that this study was undertaken. 

Purpose of the Study. The purpose of the study was 
to compute the probability that the proportion of 
hits in a single attack with a uniform pattern would 
be at least (or at most) any assigned value H. These 
computations were done for those rectangular targets 
and square patterns specified by the Army group 
which initiated the study. 


48 


PATTERN BOMBING 



0 I 2 3w^ 5 6 7 

MRE 


Figure 2. Contours for constant values of the ex- 
pected proportion E, i.e., E{H), of hits for rectangular 
patterns {Ip/ MRE X Wp/MRE) and square targets 
{It/MRE = Wp/MRE = 1). 

Method of Analysis. There is a locus for pattern 
center in the target plane, say = 0 in polar 

coordinates, such that the proportion of hits is ex- 
actly H. The probability P{H) that the proportion 
of hits will be at least H is found by numerical inte- 



Figure 3. Contours for constant values of the stand- 
ard deviation <r^ of the pro])ortion of hits, for square 
patterns (side = Ip/ MRE) and square targets (side = 
Ip/MRE). 

gration within this contour which leads to the fol- 
lowing formula: 

P[H) = j (1 - . (7) 

i 

Here A 0 is the interval in 6 over which R is treated 
as a constant and Ri is the value of R corresponding 
to the fth value in the set of ^’s. 


Simpler formulas apply in certain cases. 

Results. The results were presented in charts of 
the form shown in Figure 4. Several values read from 
the curves, together with the expected proportion 



Figure 4, Probability that in a single formation 
attack the proportion of hits will be at most (or at least) 

H. Square pattern of area 5 X 10® sq ft and target 
800 X 800 ft. 

E{H) of hits, were collected in a small table associ- 
ated with each graph. Calculations have been ex- 
tended to the sets of conditions which are itemized 
in Table 1. 

Sets of tables and graphs and a full discussion of 
the theory are contained in a report^ by AMP. 


*4 probability distribution of the 

PROPORTION OF COVERAGE WITH 
UNIFORM PATTERNS 

The preceding study was concerned with the pro- 
portion of the pattern which falls on the target, the 
common criterion of success in World War II in 
bombing with high explosives. The present study 
was concerned with the proportion of the target cov- 
ered by patterns, a subject which comes to the fore 
in toxic-gas bombing, or in bombing with any other 
area-covering weapon. 

Purpose of the Study. The purpose of the study was 
to estimate the number of attacks, with specified 
aiming-error distributions and patterns, needed to 
give a probability of at least P that at least the pro- 
portion F of the target would be covered at least 
m times. 




rL_ 


PROBABILITY OF PROPORTION OF COVERAGE 


49 


Method of Analysis. The method of analysis con- 
sisted of a model experiment in which a series of syn- 
thetic random-bombing operations were performed, 
with enough replications to permit the estimation of 
probability levels from order statistics. The data, so 

Table 1. Values of the parameters used in the study of 
the distribution of the proportion of hits, for uniform 
patterns. All 480 combinations of the values of the 
parameters were considered. 


Targets 

Square patterns 
(area) 

CEP's 

1 X 1 

1 X 3 

25 

4 

2X2 

1 X 5 

50 

8 

4X4 

2X5 

100 

12 

8X8 

2 X 10 

150 

16 

16 X 16 

2 X 50 

300 




500 




750 



accumulated, was then used as the basis for an em- 
pirical function whose general properties were sug- 
gested by theoretical considerations. 

This work was brought to a hurried conclusion at 
the end of the war. The empirical formula obtained 



Figure 5. Typical scatter-chart showing computed, 

Sc, vs observed, So, values of s, the number of attacks 
needed to make the probability at least P that the 
proportion of /n-fold coverage will be at least F. 

met most of the theoretical conditions of adequacy, 
though it did not satisfy all of them, and fitted the 
data satisfactorily, as evidenced by the typical sam- 
ple of observed versus computed values plotted in 
Figure 5. Because the empirical formula fails to 


meet all the theoretical conditions, it is not safe to 
apply the formula indiscriminately outside the ob- 
served range. 

Results. The principal result of the study is the 
following rather formidable formula: 

4F - 5 log (1 - P) + (m - 1) (2 -b 5P) 

^ 5p2‘ Pg-^i + P(1 + 2P) te-^^ 

(8) 

where 



+ {2F-1) t [8(1-F) F-5(9-5F) «+20(4-3F)] 
(80 

+ H (22P2 -23P + 5)+ (4< -9)' | vT 

( 11 ) 

Here p,t = side of pattern and target expressed in 
terms of the mean radial error MRE as unit, 

F = least proportion of target covered, on the 
average, 

m = number of times the fraction F of the target is 
covered, 

P = least probability that the least coverage is P, 
and m-fold, 

s = number of patterns required in order that 
the probability will be at least P that the 
m-fold coverage is at least F. 

Sets of charts, of which Figure 6 is typical, are 
available. In these, equation (8) is mapped for almost 
all values of the arguments used in deriving it, and, 
consequently, for values of the arguments for which 
it can be guaranteed to produce good values of s. 
These values are : 

F = 0.2, 0.5, 0.8 
P = 0.2, 0.5, 0.8 
m = 1, 2, 3, 4 

t = 1, 2, 3, 4 (and 0 when m = 1). 

The charts referred to, a detailed discussion of the 
theory, and all of the original data are contained in 
an AMP memorandum.® 


50 


PATTERN BOMBING 


4 5 STATISTICALLY UNIFORM PATTERNS 

In the preceding sections it has been assumed that 
the bomb pattern is a perfectly uniform area. In this 
study the assumption was modified ; it was recognized 
that a pattern consists of a certain number N of 
bombs each having a radius R of effectiveness, and 
it was assumed that they constituted a sample from 
a statistically uniform distribution. 

Purpose of the Study. The purpose of the study was 
to estimate the probability P'ks that in s attacks there 



0.1 .2 .3 .4 . 5 . 6 . 7 . 8.9 1 2 3 4 5 678910 

P 


Figure 6. Chart of number s of attacks vs side p of 
square pattern needed to make the probability at least 
P = 0.8 that the proportion of single coverage will be 
at least 0.8, for square targets of side t; all lengths are 
measured in terms of the mean radial aiming error 
MRE as unit. 

would be exactly k hits (1) on a sub-target of radius 
R, or (2) on a point sub-target by bombs of effective 
radius R. Items (1) and (2) are alternative state- 
ments of the same mathematical problem. 

Method of A nalysis. The method of analysis is that 
of formal probability theory. The probability PL 
of making exactly k hits in s attacks on a target at 
the point (x,y), may be written 



where, for small values of R, 




G is given by equation (6). 

Perhaps the quantity 

Pis = 1- P'os , (14) 

the probability of at least one hit, is of greatest 
interest. 

Results. The formulas of the preceding paragraphs 
were applied to the problem of finding the optimum 
pattern dimensions, i and ^ (measured in terms of 
(Ta, standard deviation of the aiming-error distribu- 
tion), for a knock-out attack on a target comprising 
many important sub-targets. The criterion is that the 
probability defined in equation (14) of the destruc- 
tion of a sub-target [at one corner ix',y') of the target 
area] be large. A nomogram is provided^ for the esti- 
mation of L and w- 

Similar formulas were applied to the problem of 
determining the pattern, L* X W* (measured in 



F'igure 7. Graph showing the probability that the 
proportion of hits will be at most (or at least) H plotted 
against H. For single attacks (s = 1); for operational 
patterns of area 3 X 10^ sq ft and circular target of 
radius 500 ft. Circles indicate observed points. 

terms of aa, standard deviation of aiming-error dis- 
tribution), which is optimum in the sense that it 
maximizes the expected number of sub- targets which 
may be hit in a single attack on a target area com- 
prising many uniformly distributed sub-targets. A 


PROPORTION OF HITS, PATTERN AREA, AND MRE 


51 


nomogram is provided for the estimation of L* 
and 

This work is discussed at length in two AMP 
papers'^ prepared by the Statistical Laboratory of 
the University of California. 

^ 6 dependence of proportion of hits 

ON PATTERN AREA AND MRE AS DETER- 
MINED BY OPERATIONAL PATTERNS 

In the studies of the preceding sections attempts 
have been made to set up simple working models 
which would serve as satisfactory bases for pattern 
bombing theory. Comparisons between theory and 
practice indicate that even the simplest model is 
often quite successful. 


However, there are certain difficulties in the way 
of bringing theory and practice into complete accord, 
simply because the physical situation is much more 
complex than a tractable, simple model. The present 
study sought to overcome some of these difficulties by 
letting data from a large number of past operations 
tell their own story, assisted very little by theory. 

Purpose of the Study. The purpose of the study was 
to determine the probability distribution of the pro- 
portion H of hits, for specified values of the mean 
radial error (Gaussian distribution) and pattern area, 
on the basis of operational reports regarding the ob- 
served proportion of hits, pattern area, and indi- 
vidual values of aiming error. In this study one must 
watch carefully in order to distinguish between mean 
radial (aiming) error and aiming error; the distinction 
is crucial to an understanding. 


Table 2. Percentiles of at most (at least) the proportion H of hits, the expected proportion E{H) of hits, and the standard 
deviation ch of the proportion of hits for operational pattern of area 3 X 10® sq ft and circular target of radius 500 ft. 


Number 

of 

attacks 

MRE 

feet 

Percentiles of at most H hits 

E{II) 

(Th 

10 

25 

50 

75 

90 


0 

0.17 

0.25 

0.35 

0.44 

0.52 

0.35 

0.14 


500 

0.07 

0.15 

0.25 

0.35 

0.44 

0.25 

0.14 


1,000 

0.00 

0.03 

0.14 

0.25 

0.36 

0.16 

0.15 

1 

1,500 


0.00 

0.02 

0.15 

0.30 

0.09 

0.13 


2,000 



0.00 

0.07 

0.24 

0.06 

0.12 


2,500 




0.00 

0.18 

0.04 

0.10 


3,000 





0.12 

0.03 

0.08 


0 

0.22 

0.28 

0.35 

0.41 

0.47 

0.35 

0.10 


500 

0.13 

0.18 

0.26 

0.32 

0.39 

0.25 

0.10 


1,000 

0.03 

0.09 

0.16 

0.23 

0.30 

0.16 

0.11 

2 

1,500 

0.00 

0.01 

0.07 

0.16 

0.23 

0.09 

0.09 


2,000 


0.00 

0.02 

0.11 

0.19 

0.06 

0.08 


2,500 



0.00 

0.07 

0.15 

0.04 

0.07 


3,000 




0.04 

0.11 

0.03 

0.06 


0 

0.26 

0.30 

0.35 

0.39 

0.43 

0.35 

0.07 


500 

0.16 

0.21 

0.25 

0.30 

0.34 

0.25 

0.07 


1,000 

0.07 

0.11 

0.16 

0.21 

0.25 

0.16 

0.08 

3 

1,500 

0.01 

0.04 

0.08 

0.13 

0.18 

0.09 

0.06 


2,000 

0.00 

0.01 

0.05 

0.09 

0.14 

0.06 

0.06 


2,500 


0.00 

0.02 

0.07 

0.11 

0.04 

0.05 


3,000 



0.01 

0.05 

0.08 

0.03 

0.04 



90 

75 

50 

25 

10 



Percentiles of at least H hits 



52 


PATTERN BOMBING 


Method of A nalysis. In single attacks (s = 1) with 
the pattern, the target, and the aiming-error distri- 
bution all specified, the probability that the propor- 
tion of hits will equal or exceed the fraction H may 


be written as the Stieltjes integral 



(15) 

Here 


Q(r) = r -2 dx 

Jo <^a 

(16) 


and GiHir) is the probability that, given a specific 
value of the aiming error r the proportion of hits will 
be less than H. GiHjr) is determined exclusively from 
the data. 

Similar formulas are developed and evaluated for 
s = 2 and s = 3 attacks, and approximations for 
use with higher values of s are explained. 



Fioure 8. Graph showing the probability that the 
proportion of hits will be at most (or at least) // plotted 
against H. For double attacks (s =2); for operational 
patterns of area 3 X 10® sq ft and circular target of 
radius 500 ft. 


Residts. The principal results are displayed in a 
set of tables and charts, of which Table 2 and Figures 
7, 8, and 9 are typical. From these can be read the 
answers to almost any problem involving the propor- 
tion of hits in pattern bombing. 

The tables and charts referred to above, and a de- 
tailed discussion of the procedure, appear in a re- 
port® by AMP. 


i 7 DEPENDENCE OF PATTERN AREA AND 
MRE ON OPERATING FACTORS AS DETER- 
MINED BY OPERATIONAL PATTERNS 

The preceding study went a long way toward es- 
tablishing quantitatively the relationship between 
proportion of hits on the one hand and the pattern 
area and MRE on the other, but it did not indicate 



H 

Figure 9. Graph showing the probability that the 
proportion of hits will be at most (or at least) H plotted 
against H. For quadruple attacks (s = 4); for oper- 
ational patterns of area 3 X 10® sq ft and circular area 
of radius 500 ft. 

the roles played by operating factors. In the present 
study that was attempted. 

Purpose of the Study. The purpose of the study was 
to discover quantitative relationships between pat- 
tern statistics (length, width, area, etc.) and mean 
radial error on the one hand, and various operating 
factors (altitude, aircraft type, numbers of bombs 
and aircraft, etc.) on the other. 

Method of Analysis. Regression equations were set 
up and solved using all of the suitable data compiled 
during the last year’s operations of the Eighth Air 
Force in ETO. 

Results. The principal results of the study are a 
series of regression equations. The coefficients in the 
regressions for various pattern statistics are given in 
equations (17) to (24) displayed in Table 3, and a 
regression for mean radial aiming error follows : 

\ogio{MRE) = 0.514 + O.OllST + 0.00573A 
+ 0.278 logio(lOOLF) - 0.0672Ri + O.I 6 IR 2 
H- O.O 2 OOR 3 + 0.00747J5 + 0.364/(0. (25) 


CLUUiTniiTfiTTI \ h 



PATTKRN AREA, MRE, AND OPERATING FACTORS 


53 


'Pable 3. Regression ecjiiations for various i)attern statistics, based on operational patterns. 


Coefficients of 


Eqiwtion 

Regression 

I 

T 

V 9 

V 12 

V 18 

A 


(AVD^ 

t 


t cos "(^-4) 

Remarks 

(17) 

'-0 

0.0452 


— 0.45o 

—0.325 

-0.164 

0.0229 

0.00670 


0.0515 

—0.00287 


B-17’s 

(18) 


—0.0225 


—0.342 

—0.224 


0.0278 

0.00576 


0.0604 

—0.00409 


B-24’s 

(19) 



0.122 

—0.459 

—0.295 

—0.167 

0.0242 


0.216 

-0.00656 


—0.00633 

B-17’s, B-24’s 

(20) 

logic ^ 


0.059 

0.668 

0.733 

0.788 

0.0143 


0.155 

— 0.00355 


—0.00227 

B-17’s, B-24’s 

(21) 

login W 


0.060 

0.876 

0.975 

1.048 

0.0098 


0.061 

—0.00314 


—0.00402 

B-17’s, B-24’s 

(22) 

/L*\V*\ 


0.139 

—0.761 

—0.590 

—0.402 

0.0259 


0.192 

—0.0118 


—0.00148 

B-17’s, B-24’s 

(23) 

login 7.* 


0.077 

0.481 

0.560 

0.645 

0.0153 


0.158 

—0.0070 


0.00070 

B-17’s, B-24’s 

(24) 

login W* 


0.058 

0.761 

0.850 

0.953 

0.0105 


0.035 

—0.0048 


—0.09218 

B-17’s, B-24’s 


In these equations the symbols have the following 
meanings: L, W, L*, Ib*are pattern length and width 
expressed in hundreds of feet, measured according to 
two criteria: L, W are measurements which include 



0.2kL 

•0.2 0 0.2 0.4, _ 0.6 0.8 1.0 


I I LJl_j 1 I I 

I 2 4 6 8 10 

Pr 

Figure 10. Graph showing the relationship between 
observed average values of log P and values*of log Pc, 
i.e., logio (LlF/100), computed from the regression 
equation (19). 


is expressed in millions of square feet. / = 0 or 1 for 
salvo or minimum intervalometers. T = 0 or 1 for 
B-17 or B-24. iVg, Nu, = (1,0,0), (0,1,0), or 
(0,0,1) according as the formation is standard for 
9, 12, or 18 aircraft. A = altitude in thousands of 
feet. Nb = number of bombs per aircraft. Bi, B 2 , 
B 3 — (1,0,0), (0,1,0), (0,0,1) or (0,0,0) according as 
the order-over-target is 1, 2, 3 or > 3, while B is 
order-over-target without qualification over the 
range 1 ^ B ^ 9; t = months, beginning with April 
1944 where ^ = 4; and f{t) is an arbitrary function 
of t to which seven values were assigned, based on 
graphs of the data. 

Some notion regarding the goodness of fit of the 
equations to the data may be gained from Figures 10 
and 11, where the observed values of pattern area and 
MRE are compared with those computed from equa- 
tion (19) and equation (25). 

It is not practicable to discuss the results ade- 
quately here. The reader is referred to an AMP 
paper® for a full discussion. The data used are in- 
cluded therein. 

i s PRACTICE PATTERNS WITH 
CONTROLLED MISSILES 


roughly 90 per cent of all bombs in the pattern; The preceding studies of operational patterns were 
L*, R'* include 80 per cent of the bombs in range not based directly on the actual patterns, but on 
and 80 per cent in deflection, independently. MRE is certain measurements made on those patterns. In a 
expressed in hundreds of feet. Pattern area {LW/ 100) relatively few instances it has been possible to make 



54 


PATTERN BOMBING 


a detailed, accurate bomb plot of actual patterns, 
both in combat and in practice bombing. Several 
studies of these were in progress, but were aban- 
doned, at the end of the war. However, the following 
small and highly specialized study was one of this 
type completed by AMP. 

Purpose of the Study. The purpose of the study was 
to estimate the probability that the proportion of 



I03 MREjj 

\ ! ! ! 

4 6 8 10 

MREg 

Figure 11. Graph showing the relationship between 
observed average values of logio MRE and values of 
logic MREc computed from the regression equation (25). 

hits would exceed H, if the bombing were done in 
formation and with the controlled missile AZON. 

Method of Analysis. Practice patterns of AZON and 
others of standard bombs were used in a model ex- 
periment. The pattern plots were dropped (figura- 
tively) some hundreds of times at points determined 
by a table of random deviates from a Gaussian dis- 
tribution. It was assumed that the standard bombs 
were affected by an error distribution in which 
MRE = 600 feet and that the AZON patterns were 
similarly affected but that the deflection component 
of the error was reduced effectively to zero. 

Results. The results are displayed in a series of 
graphs of which Figure 12 is a typical example. Here 
the probability P{H) that the proportion of hits will 
exceed H is plotted against H. The three curves cor- 


respond to standard bombs, AZON excluding failures, 
and AZON including failures. 

The method and results are given in full in a 
report^^ by AMP. 


49 SYNTHETIC PATTERNS FOR 
CLEARANCE OF MINEFIELDS 

A few studies have been carried out in which an 
attempt was made to synthesize patterns by putting- 
together a geometrical array of trains, sometimes 
taking into account the variations in spacing between 
aircraft, in release times, etc. Some work of this kind 
has been done analytically, but it seems that when 
it is considered as a standard method for analyzing 
bombing problems, the rewards are not great enough 
to compensate for the labor. It has, however, served 
usefully as a verification of the approximate ade- 
quacy of simpler theory, such as that of statistically 
uniform patterns. 



H 

Figure 12. Probability P{H) that the proportion of 
hits will be at least // vs the proportion H of hits, for 
standard bombs (solid), AZON excluding failures 
(broken), and AZON including failures (dotted); for at- 
tacks on circular target of radius 250 ft, from an altitude 
of 15,000 ft, with a 30-bomb pattern dropped by a for- 
mation of eight aircraft. MRE = 600 ft normally, but 
(Tad = 0 for AZON. Vertical bars indicate positions of 
the means. 

In the following studies such problems were solved 
by statistical experiments, or by graphical methods. 
They are discussed briefly here because of their 
value in bombing research. The methods are given in 
greater detail in Chapter 7. 




SYNTHETIC PATTERNS FOR A MANEUVERING TARGET 


55 


Purpose of the Study. The purpose of the study was 
to determine the number of heavy-bomber forma- 
tions which must attack a minefield in order that the 
probability of clearing a proportion of a path be 
0.5 or 0.9. 



0 20 40 60 80 100 

F 


Figure 13. Number, 6s, of aircraft attacks with 
6-aircraft heavy-bomber formations vs proportion F of 
best path cleared. Probability, P = 0.9, that clearance 
is at least F. Minefield 6 X <» , <ra = 6, = 0.3, / = 0.3, 

path width = 0.3, n = 12. Dispersion of the train com- 
ponents of the pattern measured by at = 3.4. 

Method of Analysis. The problem was solved using 
a model experiment. Fifty synthetic train stencils 
(incorporating bomb dispersion) were prepared. A 
set of six stencils, selected at random, was placed 
on a map of the minefield in such a manner that the 
center of the set would fall on a mark indicating an 
aiming error drawn from a known Gaussian distri- 
bution. The process was continued, with periodic in- 


spections to determine the proportion F of clearance 
achieved along the best path; the radius of clearance, 
depending both on mine and bomb, is involved in 
this calculation. The complete experiment was repli- 
cated 30 times for each set of conditions. 

Results. The principal results are displayed in 
graphs of which Figure 13 is typical. Here, for given 
conditions (aiming-error and bomb-dispersion dis- 
tributions, number of bombs in train, width of path 
and of minefield) there is given a plot of proportion 
F of clearance along the best path versus number of 
aircraft, with radius of clearance as the family 
parameter. 

The study includes similar results for train bomb- 
ing. A full discussion of this problem appears in an 
AMP report.^® 

4 10 SYNTHETIC PATTERNS FOR A 
MANEUVERING TARGET 

So far as we know, no really satisfying analysis has 
been made of formation attacks against maneuvering 
targets. The following study, which is simply explor- 
atory, relates to the secondary activity of anti- 
submarine patrol bombers, which must be prepared 
to congregate and mount attacks on enemy surface 
ships. 

Purpose of the Study. The purpose of the study was 
to discover the spacing of aircraft and bombs and the 
direction of attack which will maximize the prob- 
ability Pi of at least one hit on a small warship; the 
attack being delivered by a small formation compris- 
ing five or six aircraft, each carrying eight bombs. 

Method of Analysis. The standard deviations of the 
aiming-error distribution were assumed to be four 
and five times the target width; since a ship with 
beam of 50 ft is large for this problem, the standard 
deviations, da, are not greater than 200-250 ft. These 
values were chosen early in the war before more 
realistic (larger) estimates came to hand. 

The analysis was largely graphical and controlled 
by the following minimax principle: There is some 
course of action, i.e., maneuver, open to the target 
which, for any specified form of the attack, will mini- 
mize the probability of at least one hit. That attack 
is judged to be best which maximizes this minimum 
probability. This is illustrated in Figure 14 where the 
probability Pi is plotted against a scale showing pos- 
sible positions of a destroyer 30 sec after a decision 
to maneuver — A and A' correspond to hard left and 
right turns, B to no turn. The spacing I referred to 


56 


PATTERN BOMBING 


on the curves is the lateral spacing between aircraft. 
According to the criterion adopted, the curve for 
spacing I = identifies the best tactic shown 



Figure 14. Probability Pi of hitting a maneuvering 
target, in a fore-and-aft attack by a 5-aircraft forma- 
tion with lateral spacing I between aircraft, vs possible 
target positions. B corresponds to target remaining on 
original course, A (or A') to a hard left (or right) turn. 

because the lowest point on this curve is higher than 
the lowest point on the others. 

Results. Perhaps the principal results of the study 
are qualitative. For example, it seems quite clear that 
some types of attack are very much better than 
others, and that it is a reasonable undertaking to 


isolate the better ones. Also, that even in problems 
such as this one, where the enemy has a great deal of 
choice in the matter of defensive countermeasures, 
attacks can be designed in which the probability of 
success is very stable and depends little on the 
countermeasures. 

In the present case, the best attack discovered was 
a beam attack, with lateral spacing of twice target 
width between aircraft, each aircraft (in effect) aim- 
ing in range so as to place its train center on the theo- 
retical locus of possible ship position ; spacing in train 
is taken as 1.5W. 

The work is discussed at length in a report^ ^ of 
the AMP. 

4 11 PHOTOELECTRIC ANALYSER FOR 
SYNTHETIC PATTERNS 

Throughout this volume — indeed as recently as the 
study in Section 4.9 — the reader will have encoun- 
tered the model experiment, used as a means to solve 
certain bombing problems which would proceed tedi- 
ously if approached by numerical integration. The 
present study concerns a device which mechanizes 
the work of a model experiment. 

Purpose of the Study. The purpose of the study was 
to design an instrument, the Photoelectric Analyser, 
which, by mechanizing the procedures of a model ex- 
periment, would quickly estimate the probability of 
at least one hit or, alternatively, the expected pro- 
portion of hits, in formation attacks on irregular 
target areas. 


GROUND GLASS 
SCREEN 



Figure 15. Diagram of Photoelectric Analyser. The principal function of the instrument is to estimate the proportion 
of hits on an irregular target. 





PHOTOELECTRIC ANALYZER FOR SYNTHETIC PATTERNS 


57 


Method of Analysis. The method of analysis was 
to measure, with a photoelectric receiver, the light 
from a ground-glass screen which is illuminated as 
follows. A white image, on a black background, of a 
synthetic bomb pattern was projected on the ground 
glass after passing through a diaphragm-stop cut out 
in the form of the target. Thus the screen was only 
illuminated by an image of that part of the bomb pat- 
tern which intersected the target. The light from the 
screen was focused on a photoelectric cell which was 
instrumented so as (1) to add the effect of successive 
images of bomb patterns, or (2) to count the cases 
which were not blank. A movie projector and a film 
with 1,500 frames were used. Each frame carried a 


picture of the bomb pattern with its center displayed 
to represent a random deviate from a Gaussian dis- 
tribution. 

Results. The study has resulted in the design and 
construction of a simple photoelectric device which 
vastly expedites the estimation of the expected pro- 
portion of hits, as well as the probability of at least 
one hit, on irregular target areas. The photoelectric 
device is shown diagrammatically in Figure 15. 

The construction and use of the photoelectric an- 
alyser is discussed in detail in a document written 
by one of AMP groups. 

Shortly before the war ended it was planned to con- 
struct several of these instruments at Wright Field. 


Chapter 5 

FURTHER INVESTIGATIONS 


51 INTRODUCTION 

T he present chapter comprises investigations 
which, for one reason or another, do not seem to 
fit neatly into the classification adopted for deter- 
mining the contents of the preceding chapters. This 
is evidence that the classification has its weaknesses, 
and in no way reflects on the importance of the 
studies discussed here, relative to those discussed in 
earlier chapters. 

As an example of this difficulty, consider the study 
Incendianj Bomb Attacks on German Targets discussed 
in Section 5.5. It is concerned with incendiary at- 
tacks on German targets and draws its information 
from American and British attacks which featured, 
respectively, formation bombing, in which all the 
aircraft of a group released on the leader, and train 
bombing, in which each aircraft sighted independ- 
ently. Thus, the study cuts squarely across the funda- 
mental classification. Further, one of the principal 
objects of the study is to determine the vulnerability 
characteristic of fire divisions. So far as it is con- 
cerned with target vulnerability, the study is almost 
unique in AMP bombing work. For these reasons 
its review has been deferred to the present chapter. 

The first 'few studies discussed immediately below 
relate to problems in which the mathematical model 
used for calculation is not necessarily a conscious 
idealization of one or another specific operational, or 
tactical, technique, that is, they are not offered as 
solutions to problems in which the tactics have been 
specified. Rather, the spirit of the approach is this : 
Here is a geometrical model which obviously bears a 
relationship to problems, or parts of problems, some- 
times met in bombing investigations. Any time the 
tactics being considered promise to give rise to ap- 
proximately this geometrical situation, and, further, 
when the probability statements made in connection 
with the situation are germane to the problem at 
hand, the results of the study may be applied. It is 
perhaps unnecessary to add that this kind of study 
is often one in which the mathematical work proceeds 
quite smoothly, or in which the results can be pre- 
sented concisely, for these often constitute the mo- 
tivation for the model. It should not be inferred that 
the studies under discussion did not arise in response 


to specific problems, for they did; but the geometry 
of a military problem can sometimes be discussed be- 
fore the tactics are selected. 

52 CONDITIONAL PROBABILITY OF 
MISSING AT MOST r SECTIONS OUT OF n 
SPECIFIED SECTIONS FOR STATISTICALLY 
UNIFORM DISTRIBUTIONS 

The present study was one of the first by AMP to 
be aimed at the problems of saturation bombing and 
the clearance of minefields by aerial bombardment. 
It provides the answers to several questions which, 
while not the most important ones in very many 
bombing situations, throw some light on several im- 
portant problems. 

The underlying assumption on which the study is 
based is that bombs are distributed over a region, 
called the target area, in the random manner asso- 
ciated with the term statistical uniformity, i.e., if 
the target area is subdivided into specific sections or 
cells of equal area, each bomb is as likely to fall in 
one section as in another. A variation of this state- 
ment, useful on occasion, is that with almost any 
distribution one may subdivide the target area into 
sections having equal probability of being hit, in- 
stead of into sections having equal area, without 
affecting the mathematical formulation. 

Purpose of the Study. The purpose of the study was 
to answer the following questions, and to mechanize 
the solutions. 

1. What is the expected number E(M) of sections 
missed, provided N bombs hit a target area compris- 
ing n specified sections? 

2. What is the probability P{M ^r) that the 
number of missed sections M will not exceed r? 

3. What number of bombs N is needed in the target 
area in order to achieve a specified probability, 
P(0 ^ r), of hitting every section? 

The emphasis is on large values of n and small 
values of r. 

Method of Analysis. An exact expression for E{M), 
the expected number of missed sections, is given by 

E(M) = 71^1-0""; (1) 


58 


' ( i rm.^jxnFj'NTi ' ATi 


PROBABILITY OF MISSING r OUT OF n SECTIONS 


59 


the variance (Tm of the number M of missed sec- 
tions is 

An exact solution for P{M ^ r), the probability 



Figure 1. Probability, P{M ^ r), of missing at most 
r sections out of n = 100 sections vs the number N of 
bombs. 


that the number of missed sections M will be at most 
r, is given by 

p(Msr) = 1 - f 1 - . (3) 

j=r 

An approximate solution for N, the number of 
bombs needed to achieve a specified probability, 
P(0 ^ r), of hitting every section, is given by 

1 

N = — n loge [1 — P'" (0 ^ r)] . (4) 

If the number n of sections is large and if the num- 
ber N/n of bombs per section is greater than three. 


then equations (1), (2), and (3) may be replaced 
by the following approximations: 

E{M) « m (5) 

<t!i « m(\ - ( 6 ) 

r 

P(M Sr) , (7) 

J=0 

where 

_N 

m = ne " . (8) 

Results. The results of the study have been tabu- 
lated in ten charts, based on the exact expression, 
equation (3) which show the dependence of the prob- 
ability, P{M ^r), that at most r sections will be 
missed, as a function of the number N of bombs in 
the target area. Each chart contains curves for 
r = 0, • • • , 4. The parameter running from chart to 
chart is n, the number of sections in the target area, 
which takes the values 10, 20, 50, 100, 200, 500, 
1,000, 1,500, 2,000, and 5,000. A typical chart is 
reproduced in Figure 1. 

When equation (7) is a satisfactory approximation, 
i.e., when N/n>S and the number n of sections is 



Figure 2. Probability, P{M ^ r), of missing at most 
r sections out of n sections vs N/n — loge u, where N 
is the number of bombs. The auxiliary scale converts n 
to loge u. 

sufficiently large, say greater than 20, then P{M ^r) 
is a function only of the variable m, and the above 
set of charts may be replaced by a single chart. This 
chart, shown in Figure 2, is drawn with the logarithm 
of minus m as abscissa : 

N 

loge ( — m) = — “ logeU. (9) 


60 


FURTHER INVESTIGATIONS 


An auxiliary scale facilitates finding loge n from n. 

Both equations (6) and (7) have been mechanized 
in a circular slide rule version under the perhaps un- 
fortunate title Area-Bombing Probabilities — unfortu- 
nate in that the unwary may be encouraged to apply 
it to a wider class of problems than is legitimate. 
This slide rule is shown in Figure 3. 

It will be observed that the answers to several 
problems may be read at a single setting of the disk 
and/or radial index; also, that it is possible to read 



Figure 3. The AMP Area-Bombing Probabilities 
slide rule. 


or set to values of n, the number of sections, and to 
values of N/n, the number of bombs per section, 
which are outside the range for which equation (7) 
is a good approximation. In this region the slide rule 
has been so calibrated as to overestimate the bomb 
requirement, N. 

A person using the graphs, or the slide rule, should 
take care to apply the results only to situations in 
which the fundamental assumption — statistically 
uniform distribution of bombs over the target area — 
is at least reasonabl}^ well satisfied. 

The formulas are developed and the charts are 
presented in an AMP report.^ 

A small number of slide rules have been manufac- 
tured for distribution to operations analysts and 
other personnel in the Services. 


5 3 PROBABILITY OF HITTING 
AT LEAST k OUT OF n SPECIFIED 
SECTIONS FOR STATISTICALLY 
UNIFORM DISTRIBUTIONS 

This study is similar in principle to the one dis- 
cussed above, but different in emphasis and in a de- 
tail or two. It is also, in a sense, an extension of the 
study discussed in Section 2.5, which was concerned 
with the slide rule for Small-Target Bombing Prob- 
abilities. 

Its motivation may be traced to the need to esti- 
mate bomb requirements on targets which contain a 
number of especially important units, and where it 
is necessary or desirable that several of these units 
receive hits. As examples, one may cite the elements 
of a German V-1 installation, the compartments of 
a ship, or the units in a battery of coke ovens. 

Purpose of the Study. The purpose of the study was 
to calculate the probability of hitting at least k out 
of n specific sections, when n is small. In contrast, 
the preceding study was concerned with the proba- 
bility of hitting almost every specified section out of 
n, when n is large. But the major difference between 
the studies lies in the fact that here the probabilities 
are not conditional, i.e., account is taken of the 
probability that the bombs hit the target area, as 
well as of their distribution over the target area pro- 
vided they hit it. 

Method of Analysis. Let p be the single-shot prob- 
ability of hitting any specified section in a target 
area comprising n sections. As in the above study, 
each section has an equal probability of being hit, 
but here np 9 ^ I since p will be assigned values less 
than Ijn. Then the probability, PiH^k), that the 
number H of sections will exceed k is determined by 

k 

PiH^k) = 1 - ^ r-jirOG-u.)- 

[I — (n — k + r)p]^ , (10) 

where N is the number of bombs. 

Residts. Values of P{H^k) have been calculated 
from equation (10) for values oi k, n = 1, 2, • • 10, 

and for p = 0.1 and 0.01. Additional calculations 
suggest that if the value of p, say p', is less than 
0.01, then 

( 11 ) 

approximately. 

It was planned to calculate equation (10) for more 
values of p > 0.01, so as to facilitate interpolation; 
however, this work ceased when the war ended. 


r!( mid.DLii j U LiI Xu: 


PROBABILITY OF HITTING ALL UNSPECIFIED SECTIONS 


61 


The results are displayed on 20 charts, of which 
Figure 4 is an example. Each chart is for a fixed value 
of n and a fixed value of p. The probability, P{H 
of achieving hits on at least k of the n sections is 



0 — 

0 40 80 120 160 200 240 280 

N 


Figure 4. Probability, P{H ^ k), that at least k 
out of n = 5 sections will be hit vs the number N of 
bombs, for the case where the probability of hitting a 
section is p = 0.01. 

plotted against the number of bombs N. The various 
curves on a chart correspond to the values of k, 
which range from 1 to n. 

The charts appear in a note^ written by AMP. 

54 CONDITIONAL PROBABILITY OF 
HITTING ALL UNSPECIFIED SECTIONS FOR 
STATISTICALLY UNIFORM DISTRIBUTIONS 

This study differs from that discussed in Section 
5.2 only in that the subdivision of the target area 
into sections is contemplated as an a posteriori event 
instead of as an a priori event. It arises, for example, 
in connection with the problem of neutralizing bomb- 
er runways so that they cannot immediately be used 
as fighter strips. 

Purposes of the Study. The purposes of the study 
were: 

1. To determine the number N of bombs, distrib- 


uted with statistical uniformity, which must hit a 
bomber runway in order to preclude, with confidence 
a, the subsequent discovery and use of an undamaged 
section suited to fighter aircraft. 

2. To determine, if possible, a rough rule-of-thumb 
by which the area-bombing probabilities slide rule 
(see Section 5.2 under Results) may be adapted to 
this problem. 

Method of Analysis. The bomber runway is of 
length L and width W and one wishes to pit it so that 
the probability will be P that no fighter strip of 
length I and width w will remain. The problem is 
simplified by assuming that possible fighter strips 
must have their sides parallel to those of the bomber 



Figure 5. Number N of bombs required to give a 
0.5 probability of eliminating all fighter strips of dimen- 
sions I X w from a bomber strip of dimensions L X W, 
plotted against l/L. 

runway. Since W is usually quite small compared to 
L, this is probably not a serious limitation. 

The problem was solved by synthetic bombing. 
The coordinates of a bomb were taken from a two- 
digit table of random numbers, the impact point so 
found being marked on a chart comprising 100 X 100 
lattice points. After each bomb was plotted the chart 
was examined to see whether all possible fighter 


62 


FURTHER INVESTIGATIONS 


strips of given dimensions had been eliminated. When 
all rectangles of a given size had been eliminated, one 
noted the number N of bombs which had been 
plotted; this was continued until the smallest fighter 
strips of interest had been eliminated. This process 
was repeated until ten charts had been prepared, 
which yielded ten observations on the decisive values 




SYMBOL ^ 
w 

O 0.2 

0.3 

a 0.4 

A 0.5 

T - 


> 



— O <3 

; i 

\ i 

1 





0.20 0.30 0.40 0.50 

I 

L 

Figure 6. Correction factor A, indicated by the 
curves of Figure 5, to be applied to the slide-rule results, 
the purpose being to widen the class of problems to 
which the slide rule is applicable. 

of N for each fighter-strip size of interest. In the 
present investigation the values 0.2, • • -,0.5 for 
IjLy and for wjW, were considered. 

If, for a given fighter-strip size, the ten observa- 
tions are arranged in ascending order, say iVi,* • •, 
A^io, then a value midway between and iVe is an 
estimate of the number of bombs required to give a 
probability P of success of 0.5. Similarly, a value mid- 
way between and A^io provides an estimate of N 
for P = 0.9. 

Values of N determined from the area-bombing 
probabilities slide rule, by entering the number of 
a priori sections, n = LW llw, and the desired value 
of the probability P of hitting every section, are 
compared with the empirically determined solutions 
to the present problem. 

Results. The study resulted in preparation of a 


set of graphs for each value of P, 0.5 and 0.9, in 
which N is plotted against l/L, ior w/W = constant, 
and a family of consistent curves is drawn. One of 
these graphs is displayed in Figure 5. 

From comparisons, of the type shown in Figure 6, 
of these values with those given by the just sug- 
gested use of the slide rule, one sees that a simple 
and accurate correction factor does not exist which 
can be applied to the slide rule results, for the factor 
depends on the value of w/W and, as a matter of 
fact, on the value of P. However, the factor usually 
lies in the range 1.5 to 2.0, and decreases as P 
increases. 

If very much application arises for this type of 
problem, the present work can afford to be extended; 
the material covered here is a small exploratory study. 

The work described above is reported as a working 
paper^ of AMP’s Bombing Research Group at 
Columbia University. 


INCENDIARY BOMB ATTACKS 
ON GERMAN TARGETS 

In connection with Army-Navy Project 23 [AN-23] 
a study was made of the fire-raising effectiveness of 
the principal incendiary munitions used against 
German industrial targets. The data, derived from 
Eighth Air Force and Royal Air Force operations, 
were rather scanty; it was necessary to select for 
study those cases in which pre-raid and post-raid 
photo coverage, as well as information from Intelli- 
gence, were unusually complete and detailed. The 
munitions studied were the 4-lb magnesium bomb, 
M50, extensively used by the British and occasionally 
by us, and the 70-lb gel-filled (30 lb of gel) bomb, 
M47, the principal fire bomb used by the Eighth 
Bomber Command. 

Purposes of the Study. The purposes of the study 
were threefold: 

1. To study several assessable characteristics of 
fire divisions, notably the linear dimensions, type of 
roof, and occupancy rating, with the purpose of de- 
termining their influence on the vulnerability of 
buildings to fire. 

2. To judge the fire-raising performance of the two 
types of incendiaries, M47 and M50, under compar- 
able circumstances of target and attack, for Eighth 
Bomber Command tactics. 

3. To determine the optimum loads of 500-lb gen- 
eral purpose [GP] bombs and M47 incendiary [IB] 




INCENDIARY BOMB ATTACKS ON GERMAN TARGETS 


63 


bombs which would cause the greatest damage to 
German industrial targets. 

Method of Analysis. The principal part of the work 
deals with the estimation of the conditional prob- 
ability Pf that a single fire bomb will start a serious 
fire — serious to structure or contents — if it strikes a 


Table 1. Definition of medium height fire division 
categories, in feet. 



Fire division width category 

Data 

Narrow 

Medium 

Wide 

1M47 USAAF 

11-12 

12-19 

20-33 

M50 RAF 

12-14 

14-18 

17-22 


Note. The medium width category is 50-99 ft. The height categories were 
made to depend on the width categories so as to avoid having empty 
cells. The low and tall height categories are defined implicitly by the above. 


fire division of a given category. The fire division, 
i.e., the smallest set of rooms within fire resistant 
boundaries, is classified according to the combusti- 
bility of the roof and to the occupancy rating, de- 
fined as the percentage of the floor area covered by 
combustible material; according to the width and 
height of the fire division, narrow, medium, or wide 

Table 2. Estimates of jpf for M47 under other-than- 

combustible roofs. 


Fire division width 

Occupancy 


HE hits 

rating 

Narrow 

Medium 

Wide 

None 

Low 

0.60 

0.10 

0.05 


High 

1.00 

0.40 

0.10 

Some 

Low 

0.00 

0.00 

0.00 


High 

0.60 

0.20 

0.05 


Note. The dividing line between low and high occupancy lies between 20 
and 25 per cent floor coverage. The cumbersome phrase “other-than- 
combustible” is used instead of “non-combustible” because the latter is 
a technical phrase which does not include “fire resistant.” 

and low, medium, or tall, as defined in Table 1; 
according to the presence or absence of high ex- 
plosive [HE] hits and to the density of the bomb fall. 

Results. The principal results of the study are 
given below. 

With regard to fire division vulnerability, the fire 
divisions under combustible roofs burn more easily 
than those under other roofs. But the results regard- 
ing combustible contents, i.e., occupancy rating, 
are mixed. The probability p/ of serious damage un- 
der other-than-combustible roofs increases markedly 


with occupancy rating, as evidenced by Table 2, 
whereas p/ fluctuates erratically with occupancy 
rating when the roof is combustible. The sugges- 
tion is made that photo interpretation and intelli- 
gence may have been inadequate in the latter case. 

The more narrow the fire division the more freely 
it burns; the effect is marked. The effect of height 
is somewhat similar, i.e., the lower fire divisions 
have a tendency to burn more freely than the taller 
ones, but there is an exception : If the fire division is 
narrow, or of medium width, and low, there may not 
be enough oxygen to support a destructive fire. Nu- 
merical results for the M47 are given in Table 3. 

1'able 3. Estimates of p/ for M47 under combustible 
roofs. 


Fire division width 


Roof 

Narrow 

Medium 

Wide 

Low 

0.50 

0.40 

0.35 

Medium 

1.00 

1.00 

0.20 

Tall 

1.00 

0.15 

0.00 


High explosive hits are somewhat beneficial if the 
roofs are combustible, and the opposite tendency is 
noted for other-than-combustible roofs. HE hits are 
beneficial in the case of the narrow, low fire division, 
where increased ventilation is needed. With regard 
to density of bomb fall, the probability of serious 
damage with a single hit in a fire division is enhanced 

Table 4. Dependence of p/ on bomb-fall density, for 
narrow fire divisions under combustible roofs attacked 
with M50 by the RAF. 


Occupancy rating (per cent) 

Bomb-fall 0-15 20-30 >30 

density 


0.98(2) 

0.00 

0.04 

0.26 

2.00(4) 

0.05 

0.39 

0.78 

2.85(3) 

0.06 

0.39 

0.97 


Note. The bomb-fall density represents the average number of IB sticks 
whose centers lie within a 700 X 700-ft square; the numbers in parentheses 
indicate the number of industrial regions included in the average. The 
values of Pf in the body of the table are averages of calculated upper and 
lower limits. 

when the density is high, indicating that events in 
neighboring fire divisions are not independent, con- 
trary to the assumption in most calculations, includ- 
ing the present ones. See Table 4. 




64 


FURTHER INVESTIGATIONS 


For formation attacks in which equal loads of M47 
or M50 are carried, it appears that these types of 
bombs produce quite comparable results. If the roofs 
are combustible, the results with M50 are estimated 
to be somewhat more favorable, and the opposite 
holds when the roofs are in the other-than-combusti- 
ble category, as may be seen from Table 5. This table 
is derived from data and calculations based on five 
targets attacked by the Eighth Bomber Command 
using M 50-type bombs. 

With regard to the optimum IB/HE mixture the 
conclusions are tentative. The position appears to 
be that pure IB attacks are generally most favorable. 

Table 5. Comparison of number of fires observed, when 
M50 is used, with number expected, when M47 is used; 
based on five targets. 


Fire Combustible roof Other-than-combustible roof 

division 


width 

M50 bomb 

M47 bomb 

M50 bomb 

M47 bomb 

Narrow 

17 

13.5 

4 

6.7 

Medium 

16 

15.0 

3 

5.1 

Wide 

4 

5.1 

3 

2.9 


However, in the case of difficult fire targets, say ones 
with wide fire divisions under other-than-combusti- 
ble roofs and having low occupancy rating, pure HE 
attacks are judged to be as efficacious as pure IB 
attacks. Mixed attacks appear to be least favorable, 
but the effects of HE on fire fighters and water mains 
have been discounted in the analysis. 

The details of the results are presented in several 
AMP papers. A number of the details of an- 
alysis, not given explicitly in the AMP papers may 
be found in two British documents,^’^ and in progress 
reports^ of the Statistical Laboratory of the Uni- 
versity of California. 

BLAST EFFECT VERSUS BOMB SIZE 

This study, undertaken at the request of the Joint 
Target Group, AC/AS Intelligence, Hq AAF is based 
on a very limited set of data relating to American 
and British bombs. 

Because the British normally used mixed loads of 
incendiary and blast bombs, it is only in the excep- 
tional case that one can confidently identify observed 
damage with blast effect. When this identification is 
reasonably certain, the fact that a score or more of 


blast-bomb types were commonly employc^d makes it 
difficult to identify the observed damage with the 
type which produced it. This latter identification was 
attempted initially by ascribing to each bomb type 
the incident of observed damage which, on the 
basis of inspection and calculation, seemed most 
likely. It later became obvious that these data should 
be abandoned when a study of the American data, 
involving a single blast-bomb type and without the 
incendiary complication, indicated very great vari- 
ability in the effect of blast. Accordingly, the 
study was confined to the American data, supple- 
mented by a handful of British data whose ante- 
cedents were fairly well established. 

Because of the limited data, greater interest may 
attach to the method than to the numerical results. 

Purpose of the Study. The principal purpose of the 
study was to estimate with reference to German 
housing the mean area of effectiveness, MAE, of 
various large blast bombs. While it was planned to 
obtain such estimates for each of the principal muni- 
tions used by the American and British Air Forces, 
paucity of data restricts the study to the American 
500-lb GP, the British 4,000-lb HC, the British 
4,000-lb M2, and the British 8,000-lb HC bomb. 
Even for these four types the quantity of data is so 
small as to preclude making very reliable determina- 
tions of the MAE’s. 

Method of Analysis. In order to measure the damage 
in each incident a transparent overlay is prepared. 
This comprises a set of concentric circles, the radius 
of the kth. being proportional to k, and several sets 
of radial line segments, 6k — 3 in number in the kth. 
ring, which subdivide the rings into regions of equal 
area. 

This overlay is placed on a tracing of the incident, 
its center at the estimated point of burst. Readings 
are made for the fth incident, of the area of build- 
ings damaged, Yik, and of the area of buildings ex- 
posed to risk, Xik, within the kth ring. 

In this manner, for each type of bomb, two series 
of values were obtained, 

A t7o, I iA; t 1, • • ‘ f U 

A:=l,...,s (12) 

where n is the total number of incidents relating to 
the specified bomb type and s is the greatest value 
of k for which the damage, Yik, is not zero. On the 
assumption that the variance of Yik is proportional 
to Xik and to al (defined below), the best unbiased 


C' 


BLAST EFFECT VERSUS BOMB SIZE 


65 


linear estimate of the mean effective area for a given 
bomb type is 

s 

MAE = ^ Auq„ , (13) 

k= 1 

where 

(14) 


the area of the kth ring, 


n 



1=1 


and the least-square estimate of the standard error of 
MAE, IX, is determined by 



i=l 

The symbol oq is determined by 


(16) 


(i\r - .) ^ 5] (y<.- qkXi,YXf, , (17) 

k i 

where the outer summation is overall bomb types, N 
stands for the number of terms in the triple sum, and 
V stands for the number of qus in the double sum 
over k and bomb type. Zero values of o-f, Xik, and 
as well as = 1, are excluded from equations 
(16) and (17). Estimates of erf, namely, VI are 
calculated from 

{n-\)VL = ^ (F« - qtXi.yxt- (18) 

i 

The values of FJ. are plotted against k, and a hand- 
smoothed curve drawn, from which estimates of erf. 


now relatively free from sampling fluctuations, arc 
read. 

Residts. Although numerical results are obtained 
for each of the four bomb types treated, internal evi- 
dence suggests that misidentifications of bomb type 
with damage and/or unrepresentative sampling affect 
even the small sample of incidents which are finally 
retained. The 500-lb GP is the exception to this, but, 
while the estimate of MAE is believed to be unbiased 
in this case, the quantity of data is so small as to 
lead to a weak determination. Also, in the case of the 
8,000-lb HC, it is possible to estimate a lower bound 
for the MAE with which to compare the value calcu- 
lated by methods described earlier, which may be 
considered as an upper bound. 

The results for these two bomb types are given in 
Table 6. From these values, i.e., about half an acre 
of destruction per ton of bomb weight in each case. 


Table 6. Estimates of the mean effective areas MAE 
and of the standard errors for two bomb types against 
German housing; in acres per ton. 


Bomb type 

MAE 

^MAE 

500-lb GP 

0.49 

0.06 

8,000-lb HC 

0.49-0.56 

0.04 


it appears to be a matter of indifference as to which 
bomb is used. However, a very definite trend could 
in fact exist and yet escape detection in the present 
analysis. 

A detailed discussion of the data and method of 
analysis is given in a memorandum^® prepared by 
AMP. 


G 






PART II 


MISCELLANEOUS STUDIES 





Chapter 6 


TORPEDO STUDIES 


<^1 INTRODUCTION 

T he amp carried out three substantial analyti- 
cal and statistical studies on miscellaneous prob- 
lems of increasing the tactical effectiveness of tor- 
pedoes in naval warfare. 

The first of these studies was made for the Navy 
Bureau of Ordnance and dealt with the determina- 
tion of the optimum spread angles for salvos of tor- 
pedoes launched from destroyers for various ranges, 
target angles, and number of torpedoes per salvo. The 
spread angle is defined as the angle between adjacent 
torpedoes in a salvo, and the optimum value of this 
for a given range, target angle, and number of tor- 
pedoes per salvo is that for which the probability of 
at least one torpedo hit is a maximum. One of the 
principal findings in this study for the particular con- 
ditions provided by the Navy was that a 1 -degree 
spread angle for all conditions produced probabilities 
of hitting almost as large as those yielded by the op- 
timum angle in each case. 

The second study was one of comparing the effect- 
iveness of a proposed submarine-launched torpedo, 
which would automatically zigzag several times 
across the path of the target ship, with that of an 
ordinary straight-course torpedo. This work was done 
for the Navy Operations Research Group. The prin- 
cipal specific result of this study was that attacks 
with the proposed torpedo were about as effective for 
bow attacks as for 70-degree target angle attacks, 
whereas the ordinary straight-course torpedo is only 
about one-sixth as effective in bow attacks and about 
three-fourths as effective in 70-degree target angle 
attacks. 

The third study was carried out for Division 7.2, 
NDRC, and consisted of the computation of lead 
angles for aircraft torpedo attacks against maneuver- 
ing ships. Tables of lead angles were computed for a 
variety of combinations of range, altitude, and air- 
speed of attacking planes against target ships of vari- 
ous classes for different speeds, target angles at mo- 
ment of release, and directions of turning. 

In the next few sections a brief sketch of the meth- 
odology used in these three studies will be presented, 
together with a short summary of the principal re- 
sults obtained by applying the methods. 


OPTIMUM SPREAD ANGLES FOR 
TORPEDO SALVOS 

^ “ ^ The Problem 

The purpose of this investigation was to determine, 
under a variety of conditions regarding range, target 
angle, and number of torpedoes per salvo, the spread 
angles in destroyer torpedo salvos which would max- 
imize the probability of at least one hit on a non- 
maneuvering target. The lead angles for such salvos 
were determined on the principle of the Mark XXVII 
torpedo director. 

The need for such an investigation is intuitively 
evident from the following consideration. Errors are 
made in aiming a torpedo at a given target — errors 
due to failure to estimate correctly target angle, 
range, lead angle, and errors of the torpedo itself 
about its own aimed course and about its assumed 
speed. These errors all combine so that they would 
produce, in a large number of trials, a distribution of 
errors by which the torpedoes would miss the target 
(or more precisely the center of the target). Now if 
torpedoes fired in a salvo should be too closely clus- 
tered there would be too much probability of their 
all missing the target. Of course, if one torpedo should 
hit, there would be a large probability of others hit- 
ting also. By spreading out the torpedoes, the prob- 
ability of all missing the target can be reduced only 
at the sacrifice of reducing the probability of multiple 
hits. So the question arises as to how much spread is 
required to yield the greatest probability of at least 
one hit. 

The study was requested by the Navy Bureau of 
Ordnance (Navy Project NO-188) and was carried 
out under AMP Study No. 71. The principal results 
of this work were reported by AMP in two publica- 
tions.^’^ 

6 2 2 Fundamental Torpedo Triangle 

If there were no errors involved in the operation of 
aiming and firing a torpedo from a destroyer at a 
non-maneuvering target ship, the situation would be 
represented by the triangle in Figure 1 composed of 
the following three lines : 




69 


70 


TORPEDO STUDIES 


1. The range line R from point of launching to the 
center of the target at that instant. 

2. The line of run r of the torpedo. 

3. The target’s path between time of launching and 
time of hitting. 



Figure 1. Diagram representing the firing of a tor- 
pedo from a destroyer at a non-maneuvering target ship. 

The remaining symbols used in connection with 
Figure 1 are defined as follows: 

2H = length of target ship 
/3 = target angle 

Xc = lead angle for hit on center of target ship 

\b = lead angle for hit on bow of target ship 

Xs = lead angle for hit on stern of target ship 

fjL = \b — Xs = angular aspect of target ship 

Ct = course angle of target ship, measured clock- 
wise from an absolute meridian, as torpedo 
is fired 

Cq = course angle of destroyer at instant torpedo 
is fired, measured similar to Ct 
B = bearing angle, i.e., the angle measured 
clockwise from the course of the destroyer 
to the range line 

V = actual water speed of torpedo 

V = average water speed of torpedo 
Vs = speed of target ship 

k = Vs/v 

tc,tb,ts — times of torpedo run for hit on center, bow, 
and stern of target ship, respectively. 

In the treatment of the problem, all distances are 
measured in yards and times in seconds. 

If there were no errors of torpedo speed in deflec- 
tion, the lead angle for a hit could lie anywhere be- 


tween X/, and Xs. The lead angle Xc for a hit on the 
center of the target ship will be approximately 
iO^b + XJ. This angle is given by the law of sines 


sin Xc _ sin ^ 

Vs V 

or (1) 

Xc = arc sin (k sin jS) . 

Under combat conditions the angle at which the 
torpedo is fired is measured from the attacking ship’s 
axis. This angle is called the torpedo course angle, 
which is simply the sum of the lead angle and the 
bearing angle B. The target angle (3 is computed in 
the Mark XXVII torpedo director by the relation 


or 


+ /3 - 180° = B + Co 
^ = B + Co - Ct + 180° . 


( 2 ) 


The Probability of Getting a Hit with 
One Torpedo of a Salvo of Two Torpedoes 

First, let us consider the problem of determining 
the probability of a hit with one torpedo of a salvo 
of two torpedoes. Let the fire control error have 
standard deviation, a-/, and consider a fixed value, 
Fdf, of the fire control error. The fire control error 
is the error made by the torpedo director in esti- 
mating the correct lead angle. If 5 is the angular 
spread for the two torpedoes, then the angular errors 
(for the torpedo of the salvo having the larger lead 
angle) will be distributed about a mean (Xc+5/2-h 
Fay), with standard deviation a a. The torpedo speeds 
will be distributed about mean v with standard de- 
viation 

If the errors in speed and angle are assumed to be 
independent and normally distributed, then prob- 
ability 2 K^,v)dXdv that the angular and speed errors 
of a torpedo run will lie in the intervals (X,X + dX) 
and {v,v + dv) is given by 

p{X,v)dXdv = — • 

^TT (J 



[X - (Xc -f I + Fa /)]2 ^ 

<jI ^ (tI I 


d'Xdv. 

( 3 ) 


Now the bow and stern lead angles, X^ and Xs, 
depend on the speed v of the torpedo in such a way 
that the greater the value of v the smaller the values 
of Xb and Xs. Denoting Xb and Xs therefore as Xb{v) 


OPTIMUM SPREAD ANGLES FOR TORPEDO SALVOS 


71 


and \s(v), it is seen that the probability of a hit from 
the torpedo under consideration is obtained by per- 
forming the integration 



,v)dKdv , 


z 


(4) 


where z is the region in the \v plane for which 
\s{v) < X< \h{v). 

If the range R is large as compared with the length 
of the target ship and if the standard deviation of the 
torpedo speed is small compared with v, the angular 
aspect of the target ship, i.e., /x = Xb(r) — Xs(y), 
is approximately constant over the range of values 
of X and v pertinent to the problem. In fact, the 
value of /X is approximately given by 


H j2 sin ^ {k sin^ /3)| 
R I /l - sin^ (3 J 


A good approximation to the integral expression (4) 
can be obtained under these conditions. This approx- 
imation is obtained by replacing the two curves 
X = \b{v) and X = \s{v) which bound the region by 
the two parallel straight lines 


X = ^ + D (w - w) 


\ = - + D {v-v) , 


where D is the slope of \c{v) at z; = v and where 
XcW is given by equation (1) with v replaced by v, i. e., 

— 180A: sin /3 


V 1 - sin^ 0 

On the basis of the assumptions made above under 
which the angular aspect of the target ship is approx- 
imately constant, these two parallel lines are approxi- 
mately the same as the tangent lines to the curves 
X = 'Kb{v) and X = \s{v) at the points for which v = v. 
The approximation to the probability expression (4) 
under these conditions may be written as 

f ^ + D (v-v) 

dv I _ p{\,v)d\ (6) 

J -^ + D (v-v) 



which may be simplified by a relation in the \v plane 
and one integration 


P (F,^) 


1 

V 27r 


/ 


Zl - Fz 2 - L 3 


- 21 - Fz2 - I23 




(7) 


where /x 

~ 2 V(7j + 

^ (Fa “f“ (f\ 

8 

• (8) 

V(7i -f- 

0 2 4 Probability of at Least One Hit with 
a Salvo of Two Torpedoes 

The probability expressed by equation (7) is that 
of obtaining a hit by the torpedo (in a salvo of two 
torpedoes) having the larger lead angle, for the given 
fire control error F. The expression for this probabil- 
ity is written as P(F,8) to show that it is a function 
of the fire control error F and the spread angle 8. 

The probability of getting a hit with the other 
torpedo in the salvo of two torpedoes is obtained by 
replacing 8 by —8 in equation (7). 

The probability of at least one hit is 1 minus the 
probability of failing to hit with either torpedo. But 
the probability of failing to get a hit with either tor- 
pedo if the fire control error is F is [1 — P{F,8)] • 
[I — P{F, — 8)]. If we assume that fire control errors 
/ are normally distributed with standard deviation 
fff as a unit, then the probability of failing to get a 
hit with either torpedo, whatever may be the fire 
control error, is given by 

=7r [l-P(/,5)][l-P(/,-5)]d/, (9) 

V ZTT J-co 

and hence the probability of getting at least two 
hits for a given spread angle 8 is 

T2(5) = 1 - (32(5) . (10) 


2:2 = 

— 


6 2.5 Probability of at Least One Hit with 
a Salvo of n Torpedoes 

In the case of a salvo of n torpedoes, for which 
the spread angle is 8, the probability of at least one 
hit is a straightforward extension of equation (10) 
given by 

Pn(5) = 1 - QM (11) 

where 

1 7“ ZiL n 

Qn{8)=y= e 2 U[l-Pi{f,8)]df (12) 
V27ry-oo i=i 

where 

1 rzi - fz 2 - i(2i - n - 1)28 _ 

Pi(f,S)=-;= e ^dl. (13) 

V ZtJ- j, - - l(2i - n -1) 


roTrri'MJi.jn'iAh 


72 


TORPEDO STUDIES 


Standard Deviation of 
Fire Control Errors 


The quantity cr/, the standard deviation of fire 
control errors, is an important quantity in the com- 
putation of P„(5), and it can be estimated from the 
standard deviations <jb, and at, respectively, 

of the bearing angle errors, target course errors, 
target speed errors, and errors in setting torpedo 
tubes relative to indicated torpedo course angle. 
For it will be recalled from the discussion in Section 
6.2.2 that the target angle is estimated by the 
Mark XXVII torpedo director by the equation 

^ = Co + P - C + 180° , 
and if the torpedo course angle is T, then 


T = B 

= P + arc sin (k sin /S) (14) 

or 

T = B arc sin [k sin (C^ 4* P — Cr + 180°)] . 


Expanding T in a Taylor series about the true tor- 
pedo course angle To and neglecting terms of second 
order and higher, 

ST dT dT 

{T - To) = (A« — + (ACy^) — + (Ai;«) — . (15) 

Squaring this equation, averaging the squared errors, 
assuming independence of errors, and evaluating de- 
rivatives, one arrives at 




[ 


(tI\ 1 + 2/c 


cos ^ 
cos X 




cos^ jS”! 

cos^ X J 




cos^ /3 
cos^ X 




(180)^ sin^ ^ 
TT^ COS^ X 


(16) 


where at is the standard deviation of the errors in- 
volved in setting the torpedo tubes relative to the 
indicated torpedo course. 

The principal difficulty in actually evaluating a/ 
lies in the paucity of data on the errors involved in 
measuring B,Ct,Vs and torpedo tube setting. How- 
ever, some information exists in the “standards of 
proficiency” listed in a memorandum^ on torpedo 
training exercises. On the basis of this information, 
aj varies from 2°54' to 1°54' as the target angle ^ 
varies from 10° to 90°, and the average of a/ for 
all target angles is approximately 2°30'. 


6.2 7 Principal Computational Results 

A very extensive set of computations of values of 
P,X^) [see equation (11)], the probability of at least 


one hit in a salvo of n torpedoes, was carried out for 
three values of a/, namely 1°, 2°30', and 4°. For 
each of these values the value of P„(5)> was com- 
puted for the following values of parameters: 
Target speed Vs = 25 knots 
Mean torpedo speed v = 33.5 knots 
Torpedo speed error a^ = 0.7 knot 
Torpedo angular error at = 0°30' or 8.9 mils 
Number of torpedoes per salvo n = 2, 4, 6, 8, and 10 
Target angles ^ = 10°, 30°, 60°, 90°, 100°, and 120° 



Figure 2. Optimum spread angle and probability of 
at least 1 hit with a salvo of 4 torpedoes for various 
values of r (range/target length) and for standard 
deviation of fire control errors equal to 2°30'. 

Ratio of range to target length r = R/2H = 20, 
30, 40, 50, and 80 

Spread angles d = 0° to 4°, so as to include the 
optimum angle. 

The results of these computations are given in both 
tabular and graphical form in an AMP report.^ A 
typical graph is shown in Figure 2. On the basis of 
these calculations, numerous special tables and 
graphs are presented showing the effect of number 
of torpedoes, range, and standard deviation of fire 
control errors on optimum spread angles; effect of 
number of torpedoes, range fire control errors, and 
target angle on probability of obtaining at least one 
hit, and so on. 




EFFECTIVENESS OF ZIGZAG TORPEDOES 


73 


The most significant fact in connection with all 
these results is that although the optimum spread 
angle varies considerably from one set of conditions 
to another, the probability of securing at least one 
hit does not diminish much with small departures of 
spread angles from their optimum values. In fact it 
was found possible to select a single spread angle, 
namely 1°, which is relatively efficient for all con- 
ditions considered. More precisely, it was found that 
in about 80 per cent of the combinations of condi- 
tions under which P„(5) was computed the prob- 
abilities of at least one hit did not decrease by more 
than 6 per cent when the optimum angle was re- 
placed by a constant spread angle of 1°. 

All of these results apply to individual destroyer 
attacks. Brief consideration is also given to coordi- 
nated attacks of several destroyers on one or more 
target ships. The principal conclusion here was that 
effectiveness of such an attack can in general be 
maximized by having each attacking destroyer max- 
imize the probability of getting at least one hit itself. 

6 3 EFFECTIVENESS OF ZIGZAG 
TORPEDOES 

^ ' The Problem 

The problem here is to estimate the probability of 
hitting a target ship by a torpedo which would zig- 
zag back and forth across the path of a target ship 
in a prescribed manner, as compared with the prob- 
ability of hitting the target ship under the same con- 
ditions by using an ordinary straight-course torpedo. 
The proposed zigzag torpedo was considered for use 
in submarine operations against merchant ships. 

Two types of zigzag torpedoes were proposed for 
consideration: (1) an efficient (ideal) one which 
could follow an intricate path with a minimum of 
overlap of its region of effectiveness, and (2) a more 
practically realizable one which would cross the 
prospective course of the target ship three times. In 
a torpedo of type (I) no restriction was placed on 
the path of the torpedo although torpedo range and 
speed, target speed, and the standard errors of esti- 
mates of target speed and course were specified. 

^ ^ ^ Numerical Data 

The problem considered was quite specific as far 
as numerical assumptions were concerned, and al- 
though the theory could be generalized without much 


difficulty, it is convenient to consider the theory as 
specialized to the particular problem at hand. 

The data provided for the problem was as follows. 
Speed of torpedo = 30 knots 
Maximum range = 4,000 yd 
Range at firing = 1,500 yd 
Loss in time and speed of torpedo at turns 
ignored 

Estimated target speed = 10 knots 
Target length = 420 ft 

Standard deviation of estimate of target angle 
= 20 ° 

Standard deviation of estimate of range = 10 
per cent 

Standard deviation of incidental errors = 4° 
All errors assumed independent and normally 
distributed. 

In order to make standard deviations of both 
course and speed errors unity, the following units 
were used. 

1 linear unit (l.u.) = 1.688 yd 
1 angular unit (a.u.) = 20°. 

The torpedo speed is then 10 l.u. /sec; the esti- 
mated ship’s speed is 10/3 l.u. /sec; and half the 
ship’s length is 41.445 l.u. 

3 3 Method of Determining Efficient 
Unrestricted Torpedo Courses 

It is convenient to use a stationary polar coordi- 
nate system with coordinates r (length of radius 
vector) and 6 (angle measured from estimated course 
of ship). The origin of this coordinate system is the 
position of the center of the target at the moment 
of release of torpedo. We are concerned with the 
probability of hitting the ship at any point {t,6) 
at time t, where t is measured in seconds from instant 
of release of torpedo. The relevant values of t lie 
between 0 and 237. 

If ?/ denotes target speed, then, in order for the 
torpedo to hit the target at point {r,d) at time t, 
the target course must be d and the point {r,d) 
must be not farther from the origin than the bow of 
the target nor nearer to the origin than the stern of 
the target. This means that v' must satisfy the 
inequality 

v't - 41.445 41.445. (17) 

If V is the error in estimating the target ship’s speed, 
then 





74 


TORPEDO STUDIES 


Substituting inequality (17) and using the fact that 
rjl = v' \i Si hit occurs at (r,0) then, 

r 10 41.445 r 10 , 41.445 

t 3 3 ^ 

must be satisfied if a hit is to occur at (r, B ) . 

The inequality (18) means that corresponding to 
an actual path of a torpedo in the three dimensional 
space {r,B,t), there is an area of effectiveness in the 
plane of 6 and v which includes all of the points 
for which a hit would occur. This area is a belt 
whose center is {B,rlt — 10/3) and whose length 
(parallel to the v axis) is equal to 2(41.445/0- For any 
given path, the probability of hitting the target is 
the integral of (l/27r) exp [— M(0^ + v^)] over this 
belt (assuming 6 and the errors in the speed of the 
target ship to be independent and normally distrib- 
uted). This probability is conditional upon given 
incidental (angular) and range errors. Different 
errors would give a different path in r,d,t space and 
hence a different belt in d,v space. The absolute 
probability of a hit is the integral with respect to 
range and incidental errors of the product of this 
conditional probability by the error function of the 
range and incidental errors. 

Any actual path of the torpedo 
r = r{t) 

e = d{t) (19) 

determines the center of its belt in the d,v plane by 
the relations 

6 = e{t) 

_ r{t) _ 10 
t 3 

and conversely, the center of the belt in the d,v 
plane determines the torpedo path. For the equation 
of the center of the belt in the d,v plane the form is 
V = v{d) which is equivalent to 
r 10 

^ f = tie) . (20) 

But since the speed of the torpedo is constant 
(10 l.u./sec) we can write the equation for velocity 
in terms of horizontal and vertical components in 
polar coordinates, as 



Thus, equations (20) and (21) are equivalent to 
equation (19), showing that a torpedo path in 
r,d,t space determines a belt in B^v space and con- 
versely. 


^ ^ Application of the Method Described 
in Section 6.3.3 

The application of the method described in Section 
6.3.3 is very laborious when a distribution of range 
and incidental errors is used. Actually, the applica- 
tion of the method has been carried out only for a 
few special cases in which range and angular errors 
were assumed zero. In particular, work was carried 
out for a straight-course torpedo for a target angle 
of 30° and a torpedo speed of 30 knots, and for an 
unrestricted zigzag torpedo that followed a course 
described as follows. It was assumed that the torpedo 
remained on a straight course for 67 sec, then 
carried out a series zigzag involving seven turns. 


errors: b target angle 30* 



is identical with that of straight-course torpedo. Target 
angle = 30°, range and incidental errors zero. 

These turns were arranged so that the entire effective 
belt in the B,v plane consisted of a series of rectangu- 
lar strips so placed as to cover approximately a rec- 
tangle centered at the origin in such a way that there 
was no overlap nor any portion of the rectangle un- 
covered by the belt. The actual torpedo path in the 
water corresponding to this belt in the B,v plane is 
very complicated. But the important point is that 
the path was an efficient one in the sense that the 
torpedo swept the area about the ship (as the ship 
moved) in such a way as to avoid both overlapped 
sweeping and “holes” of unswept area. The prob- 
abilities of a hit for this particular straight-course 
torpedo attack and the corresponding unrestricted 
zigzag torpedo attack were 0.22 and 0.57, respec- 
tively, (assuming range and incidental errors to be 
zero). The integration of (I/27r) exp [— 3^ (0^ + i^^)] 
over the belt in each example was approximated by 
counting elements of area of a grid of a two-dimen- 
sional circular normal distribution in which the 
probability was represented by small elements of 
area — each rectangle representing probability equal 


75 


EFFECTIVENESS OF ZIGZAG TORPEDOES 


to 0.001. The integral over any belt would be the 
sum of the number of elements of area which covered 
the belt. Figure 3 shows the actual path considered 
in the example, while Figure 4 shows the belts in the 


0,v space corresponding to the actual path shown in 
Figure 3. It should be noted that the belts for the 
zigzag path consists of approximately a series of non- 
overlapping rectangles. 



ERRORS 
RANGE = 0 

incidental *0 


STRAIGHT-COURSE TORPEDO 
ZIG-ZAG TORPEDO ’ 


•OOOt 


TARGET ANGLE =30® 


o 

L 


LINEAR 

PROBABLE 

ERROR 







LINEAR 

PROBABILITY 


o 

p 

3.0 

3.5 


1.5 

2.0 

O 

o 

o 

p 

0.20 

P 

o 

0.40 

o 

1.495 

0.49 


Lf* 


I I I I I I I I I I I I I ■ I I I . I I I . I I I I I I I I I t 1 I I I I I I t I 1 I I t t I I I 


LINEAR average DEVIATION STANDARD DEVIATION 

0* t ^ <^0(^0-u*0i/*0 cn O O cn O U» 

* t ‘ ‘ ‘ ‘ ^ ‘ ‘ ‘ V ' ■ ' ■ * ' ' ' ' ' ■ ■ ■ ' ' ' < ' ■ I ■ ■ ■ ■ < ' ' ’ ' I ’ < ■ » » 


Figure 4. Belt covered in plane which corresponds to the path shown in Figure 3. 



76 


TORPEDO STUDIES 


Using this procedure, a detailed study was made 
of the effectiveness of a zigzag torpedo fired from 
target angles, 71.6° and 0° (for the same conditions 
stated in Section 6.3.2) and restricted to cross the 
prospective course of the target ship three times al- 
most at right angles. It was found that the prob- 
ability of a hit from a zigzag torpedo in the 71.6° 
target-angle attack was about 35 per cent greater 
than that of a straight-course 71.6° target-angle at- 
tack, while for the 0° target-angle (bow) attack, the 
probability is about six times as large as that for 
the corresponding straight-course attack. 

Bow attacks for a 60° saw-tooth zigzag torpedo 
path and a serpentine torpedo path made up of 
alternating semicircles were considered. These were 
found to be about half as effective as the zigzag path 
of three “almost” perpendicular crossings. 

The details of this investigation, which was made 
at the request of the Navy Operations Research 
Group, have been discussed in an AMP report.^ 

4 AIRCRAFT TORPEDO LEAD ANGLES 
FOR ATTACKING MANEUVERING 
TARGET SHIPS 

^ ' The Problem 

The computation of lead angles for ship targets 
by existing American torpedo directors has been 
based on the assumption that the target ship moves 
ahead in a straight course at constant speed. This 
fact has restricted the usefulness of such directors 
against high-speed maneuverable warships. Combat 
reports from the Pacific indicated that evasive action 
was almost always being taken by Japanese warships 
by turning at maximum speed in the tightest possible 
circle. The problem then arose as to what lead angles 
should be used for aircraft torpedo attacks on ma- 
neuvering warships of various kinds and how much 
these lead angles differed from those for straight- 
course targets. AMP undertook a study of this prob- 
lem at the request of Division 7.2, NDRC. 

^ ^ ^ Numerical Conditions of Problem 

Because of the absence of data on the maneuver- 
ability of Japanese warships, characteristics of ships’ 
turns were obtained for several typical warships 
(CL, CV, and BB) of the United States Fleet from 
the David W. Taylor Model Basin. In the case of 
each ship the path of the ship is graphed from the 
time the execute order is given for a full over rudder 


(35°) with a steady throttle. Initial speeds consid- 
ered ranged from 15 to 30 knots. In graphing such a 
path, the position of the ship is located at the origin 
at the execute order with ship’s axis coinciding with 
the y axis and turning to the right out into the 
first quadrant. 

The location of the attacking plane is expressed 
in terms of its range and target angle relative to the 
ship as shown in Figure 5. The altitude of the plane 

EXPECTED POINT 



and its airspeed are expressed in terms of a single 
parameter A, defined as the product of the torpedo’s 
time of travel in air by the difference between its air 
and water speeds. As will be seen in Section 6.4.3, 
A was chosen because, for a given location of the 
plane, all combinations of altitude and airspeed 
yielding the same value of A require the same lead 
angle. 

The following combinations of values of A and 
the range have been considered. 

1. A = 292, Range = 1,000, 1,250, 1,500, 1,750, 
and 2,000 yd. 

2. Range = 2,000, A = 300, 600, 900, and 1,200. 
For each case considered target angles were varied 
from 0° to 330° at intervals of 30°. The torpedo 
speed in water was assumed to be 33.5 knots. 

If the position of a ship on a definite ship’s char- 
acteristic curve is known, the position of the ship at 
future times is determined from the David W. Taylor 
Model Basin curves. If Xs and Xb are lead angles for 
a hit on the stern and bow of the ship, then any lead 
angle between Xs and Xi, will result in a hit. In prac- 
tice, however, aiming and running errors, and errors 
in estimating target ship’s speed and target angle, 
will cause a distribution of errors of lead angle from 
a perfect lead angle which would deliver the torpedo 
at the center of the ship at the moment of impact. 
The problem then is to obtain an optimum estimate 
of lead angle. 


AIRCRAFT TORPEDO LEAD ANGLES 


77 


6.4 3 Formula for Lead Angle 

If the position of a ship on its characteristic curve 
is known at each instant of time t we may express 
the position of its center in the x,ij plane described 
in Section 6.4.2 by means of the coordinates 
Xc{t),yc{i). In the analytical discussion of the prob- 
lem of determining lead angle, the following nota- 
tion will be used : 

U = time interval between order to turn and 
time of sighting 

ti,ts = time intervals between time of sighting 
and times of hit on bow and stern, 
respectively 

ta = time of torpedo’s fall from plane to water 
R = range, i.e., distance from point of tor- 
pedo release to center of target at U 
U = distance from point of torpedo release 
to point of hit 
jS = target angle at time 
X = lead angle for torpedo hit 
= lead angle for hits on bow and stern, 
respectively 

H(t) = ship’s heading at time i.e., the angle 
measured clockwise between ship’s axis 
and positive x axis 
h = height of plane at to (in feet) 
a = airspeed of plane at to 
w = water speed of torpedo 
c,d = coordinates of plane at to 
di, = angle measured from negative direction 
of horizontal axis through (c,d) and 
measured clockwise to U for hit on 
target bow 

^b(t),yb(t) = coordinates of bow of ship at time t 
^c{t),yc{t) = coordinates of center of ship at time t 
^s(t),ys{t) = coordinates of stern of ship at time t. 
All distances not otherwise indicated are measured 
in yards, all times in seconds, and speeds in yards 
per second. 

The situation from the time of sighting until a 
torpedo hit can be represented graphically as shown 
in Figure 6. 

If we let A = {a — w)ta, where ta = 2hlg, then 
the total distance traveled by the torpedo is A -f t^iv 
for a hit on the bow and A tsW for Si hit on the stern. 
The condition for a hit on the bow is that the dis- 
tance from c,d to the bow of the ship at time + k 
be equal to the torpedo run, i.e., 

[xb{to + tb) — c\^-{-[yb{to-f-tb) — d]^= [A -h tbw]^ (22) 
which is an equation for determining tb. A similar 


expression holds for a hit on the stern. The solution 
of equation (22) for t^ is carried out by trial and 
error and interpolation in any given case. Once the 
equation is solved for t^, then the lead angle \b is 
determined from the expression (see Figure 6) 

Xb = db — — H(to)] 

where 


^ ^ r ^^(^0 + 

dh = arc tan — — ; ^ 

L Xb{to + tb) — cj 

A completely similar procedure can be followed for 
determining the lead angle \s for a hit on the stern. 



Figure 6. Diagram showing the coordinates and 
angles used in describing a torpedo plane attack against 
a ship. 

The average of X^ and Xs in a single given case (cor- 
responding to one characteristic curve) was taken 
as the lead angle X. 

Because of the fact that initial speed of a ship is 
unknown, the lead angle thus computed may be in 
error. To compensate for this possible error it was 
assumed that the maneuver is executed in such a way 
that the speed is increased three times out of four 
and decreased one time in four. The lead angle was 
computed for the characteristic curve for the de- 
creased speed, and the two lead angles averaged in 
the ratio of 3:1. Investigation showed that the 
lead angle was not very sensitive to a variation of 
these weights from 1:1 to 3:1. 

The theory of lead angles and methods of com- 
puting them have been presented in two AMP 
reports. 

^ Extent of Applications of Procedure 
for Determining Lead Angle 

Computations were carried out for lead angles in 
attacks on maneuvering light cruisers, heavy cruisers, 
and battleships under a wide variety of conditions. 
Speeds considered ranged from 15 to 30 knots. Some 




78 


TORPEDO STUDIES 



Figure 7. Optimum lead angles for an aerial tor- 
pedo attack on a battleship, a light cruiser, and an 
aircraft carrier. Range = 1500 yd, A = 292, torpedo 
speed = 33.5 knots, initial target speed = 20 knots, and 
observed speed=20 knots. 

cases were considered in which the speed was in- 
creased after the order for executing the maneuver 
was given, and cases were considered in which the 


speed was decreased. Combinations of the quantity 
A and the range R mentioned in Section 6.4.2 were 
considered. 

The lead angle has been presented in both tabular 
and graphical form as a function of target angle. 
Figure 7 shows a typical set of curves. Although the 
discussion has been presented in Section 6.4.3 in the 
case of a right turn by the ship, the same tables and 
graphs can be used in the case of a turn to the left 
by entering on the tables or graphs a target angle 
obtained by subtracting the actual target angle 
from 360°. 

Tables and graphs are also presented showing the 
effect of each of the following factors on lead angle : 
target class, target speed, range, airspeed, and alti- 
tude. Comparisons of lead angles are made with lead 
angles computed on the assumption that the targets 
followed straight courses with constant speed. These 
tabulations and graphs have been presented in a 
report® prepared by AMP. 




Chapter 7 

STATISTICAL STUDIES IN MINE CLEARANCE 


7 1 INTRODUCTION 

AMP WAS REQUESTED to make several statistical 
studies of the effectiveness of various explosive 
devices and procedures for clearing antitank and 
antipersonnel mines. 

One of these studies was concerned with the de- 
sign and statistical treatment of the results of an ex- 
tensive experiment for testing the effectiveness of 
various linear explosive devices against several types 
of German and Japanese antitank and antipersonnel 
mines. In particular, curves were determined for the 
different mines buried at different depths, expressing 
the probability of detonation in terms of distance 
from the crater made by the explosive. This work was 
carried out in cooperation with the Land Mines Com- 
mittee, NDRC, for the Army Engineer Board. A de- 
tailed description of the statistical methods used in 
the study are given in Section 7.2. 

A second statistical study dealt with an investiga- 
tion of the extent of clearance of mines to be expected 
by using against beach minefields 120-rocket barrages 
launched by a device known as the WOOFUS. This 
study was carried out by means of a miniature ran- 
dom number experiment, in which the radius of clear- 
ance of a single rocket and the errors involved in de- 
livering the 120 rockets in a barrage were simulated. 
The principal conclusion of the study was that single 
barrages of rockets designed for the WOOFUS could 
not be expected to be effective against beach mine- 
fields. This study was carried out for the Joint Army- 
Navy Experimental Testing Board Section [JANET] 
and a description of the statistical methods used 
are described in Section 7.3. 

A third statistical study in the field of mine clear- 
ance was an investigation of the effectiveness of 
aerial bombing in clearing paths through minefields 
which could be used by tanks. This work was also 
done by model experimental methods (described 
more fully in Section 7.4) in which radius of clearance 
and errors in placing bombs on the minefield were 
simulated. The principal result of the study was the 
fact that very large numbers of bombs would be re- 
quired for clearing paths through minefields of the 
usual range of widths by this method. The work was 
done for the Army Engineer Board. 


Most of the work in the second and third studies 
was done by experimental statistical methods in 
which model experiments simulating the conditions 
of the problem were repeated a number of times. The 
theory underlying the two studies can be formulated 
analytically in terms of appropriate mathematical 
formulas but the computation that would have been 
involved in the mathematical approach would have 
been prohibitive. The experimental methods devel- 
oped are fairly simple but quite effective and the 
routine, once it is set up, can be made to operate at 
a clerical level. There are undoubtedly many other 
statistical problems of this type in military research 
which can be more effectively handled for practical 
purposes by experimental methods than by analytical 
methods. 

7 2 CLEARANCE OF MINES BY LINEAR 
EXPLOSIVE DEVICES 

During the war the Army Engineer Board carried 
out a testing program on several linear explosive de- 
vices for clearing paths through minefields. There 
were two classes of them. Devices of one class (De- 
molition Snakes M2, M2A1, and M3) were designed 
for clearing paths for tanks through antitank mine- 
fields, and devices of the other class (Carpet Roll, 
Ml Snake, Detonating Cable, and Bangalore Tor- 
pedo) were for similar use against antipersonnel 
mines. These devices ranged in length from five feet 
in the case of the Bangalore Torpedo, to 320 feet 
(of explosive) in the case of the Demolition Snakes. 

The problem involved in testing these devices 
against mines was that of getting enough informa- 
tion about mine detonation produced by the devices 
to be able to state, with a reasonably high degree of 
confidence, the percentage of a large number of 
mines, of a given type planted at a given depth at 
a given distance from the device, which would be 
detonated by the device. To make such a test di- 
rectly would require a prohibitive number of enemy 
mines or reproductions of them, even if they were 
available for such use in large quantities. The diffi- 
culty was largely overcome by developing a method 
of using the Universal Indicator Mine of the Army 


79 


80 


STATISTICAL STUDIES IN MINE CLEARANCE 



Figure 1. Universal Indicator Mine field pattern for M3 Snake. 


Engineer Board, which was essentially a small gauge 
for recording peak pressure resulting from an ex- 
plosion at any point where the gauge was placed. The 
relation between pressure measurements on this indi- 
cator and probability (or percentage) of detonation 
of any actual mine can be established on the basis of 


a few mines. Accordingly, the indicator, being avail- 
able in large numbers, could be used for studying the 
pressure field around a given explosive device at the 
moment of explosion. Also, one test with the indi- 
cators provided information about all mines which 
could be calibrated against the indicator. The prob- 


CLEARANCE OF MINES BY LINEAR EXPLOSIVE DEVICES 


81 


loin tlieroforc roducod to that of dosigiiing a field of 
ITnivcrsal Indicator Alines around the device so that 
there were enough mines at each of several distances 
and at each of several depths to obtain a fairly re- 
liable estimate of the average and variation of the 
pressure for each distance-depth combination. The 
experiment could be repeated for different soil con- 
ditions, e.g., dry or wet, and sand or clay. 

A group of experiments of this type designed by 
AAIP for testing three antitank mine Demolition 
Snakes and four antipersonnel mine-clearing devices 



0 10 20 30 40 

DISTANCE FROM CHARGE IN FEET 

Figure 2. Average Universal Indicator Mine read- 
ings for 3-in. hose. Ground — wet. Detonation — high 
order. 

were carried out at the Army Engineer Board Field 
Station at the A. P. Hill Military Reservation. In 
designing the experiment, a small amount of data 
was available from earlier preliminary experiments 
to indicate at what range of distances from the device 
the indicator mines should be planted in order to 
yield efficient results. It turned out that the indicator 



Figure 3. Expected percentage of German TAIi43 
mines cleared by 3-in. hose in dry ground. Depth of 
burial (in inches) given at ends of curves. 


mines had to be concentrated much more closely 
around the explosive device than originally thought 
necessary in order to get an efficient experiment. 
Figure 1 shows the layout of a typical experiment. 
The data from this group of experiments were an- 



Figure 4. Diagram showing “skip effect” in pressure 
curve. 





82 


STATISTICAL STUDIES IN MINE CLEARANCE 


alyzcd and curves of average readings (pressure) for 
indicator mines buried at depths of 2, 4, and 6 in. 
against distance from center of crater were obtained 
for each of the six mine clearance devices under sev- 
eral soil conditions. Figure 2 shows a typical set of 
curves of average readings. 

Enough German TMi43, Japanese J93, and Dutch 
Mushroom Top mines were available to be able to 
establish curves showing probability (or percentage) 
of detonation against pressure reading of the indi- 
cator mine. This made it possible to transform the 
curves described in the preceding paragraph to curves 
showing the expected percentage of detonation of 
TMi43, J93 or Dutch mines by each of the explosive 
devices at different depths of burial of the mines and 
for different soil conditions. Similar results were ob- 
tained for Schumines, Mustard Pots, and S mines, 
although the results were somewhat less reliable be- 
cause of scarcity of the mines for testing purposes, 
and because these mines could not be calibrated ac- 
curately with the indicator mine. Figure 3 is an ex- 
ample of curves of expected percentage of detonation 
of German TMi43 mines. 


An interesting by-product of this series of experi- 
ments was a great deal of quantitative information 
on the phenomenon of ^‘crater effect” or “skip effect” 
in an explosion of the type produced by these devices. 
Skip effect manifests itself as a dip in the curve of 
pressure (plotted against distance from explosive) 
which occurs at the edge of the crater, as illustrated 
in Figure 4. The skip effect varies of course with 
explosive device, depth of burial, soil conditions, etc. 

For complete details on the description of this 
group of experiments, together with graphs of the 
pressure curves and expected percentage clearance 
curves for the various linear explosive clearing de- 
vices, the reader is referred to a paper^ published 
by AMP. 

Only under exceptional circumstances would the 
explosion of a linear charge result in the detonation 
of 100 per cent of the mines in the crater or in a strip 
wide enough to accommodate a tank. Accordingly, 
under most circumstances there was a small prob- 
ability that a tank track would hit a mine in passing 
through a minefield along a linear charge crater. At 
the request of the Army Engineer Board, a study was 



Figure 5. Curves for various field widths showing expected percentage of tanks passing through a minefield without 
striking mines, plotted against density of mines in minefield. Effective track width is 24 inches. 



CLEARANCE OF MINES BY ROCKET BARRAGES 


83 



Figure 6. Curves showing expected percentage of tanks, of various total effective track widths, passing through a 
minefield without striking mines, plotted against density of mines when projected to front edge of minefield. (The 
label “Track Width” means “Total Effective Track Width.”) 


made of the probability of a tank, with given effective 
track widths, passing through a minefield of a given 
width without hitting a mine, or along the crater of 
a linear charge in which the density of mines had 
been reduced, without hitting a mine. Curves show- 
ing the probability of a successful crossing of a mine- 
field plotted against mine density (in number of 
mines per million square feet) were prepared for 
effective track widths ranging from 8 to 40 in. and 
for field widths ranging from 10 to 500 ft. These 
curves of which Figure 5 is an example were pre- 
sented in an AMP memorandum.^ The same results 
were presented in a slightly different form in another 
AMP memorandum^ in which density of mines was 
expressed as number of mines per yard of front of 
the minefield. This amounted theoretically to con- 
sidering all mines to be moved up to the front edge 
of the minefield, and then the density expressed as 
mines per yard. The effect of minefield width under 
this scheme showed up as increased density of mines 
per yard. Figure 6 shows the form in which these 
curves were issued. 


7 3 CLEARANCE OF MINES BY 
ROCKET BARRAGES 

Late in 1944, AMP was asked by the Joint Army- 
Navy Experimental Testing Board to make a sta- 
tistical study of the effectiveness with which 120- 
rocket barrages launched by a newly developed de- 
vice known as the WOOFUS could be used for 
breaching beach minefields. These launchers were 
mounted in a modified LCM(3). A few tests were 
made on an LCM in order to obtain information 
about errors due to roll, pitch, weave, and forward 
motion of the LCM. 

Some static tests had been made on the 7.2-in. 
Mk5 demolition heads, with which the rockets were 
equipped, to find out how effective they were against 
Universal Indicator Mines (see Section 7.2). From 
these tests and from the known relationship between 
pressure readings on the indicator mines and prob- 
ability of detonation of other mines, it was possible 
to estimate, for each type of mine, the relationship 
between the probability of detonation and distance 




84 


STATISTICAL STUDIES IN MINE CLEARANCE 


of mine from rocket at time of explosion. From this 
relationship an “equivalent radius of clearance” for 
the rocket head against any given mine buried at a 
given depth was determined. More specifically, sup- 
pose the probability of detonation of a mine at dis- 
tance r is/(r), and that the density of mines is con- 
stant. Then the equivalent radius of clearance Tq is 
defined by 

7rro = 27r / rf{r)dr . 

Jo 

Range tables for the rockets for each of two sizes 
of launching motors (2.25-in. Mk3 for barrages at 
range of 200-250 yd, and 3.25-in. Mkl for barrages 
at range of 400-500 yd) were available. The range of 
a rocket varies, of course, with the angle of elevation, 
and for a given motor there is inherent rocket dis- 
persion in both range and deflection from one firing 
to another. The 120 rails in a WOOFUS launcher are 
set at angles of elevation varying from 25° to 45°. 
The statistical problem may now be briefly stated 
as follows. How can one combine the effects of the 
various factors which affect the accuracy with which 
a barrage is launched at a given area on the beach 
and simulate a launching enough times to determine 
reliably what fraction of the mines will be cleared on 
the average with one barrage, two barrages, three 
barrages, etc.? The area at which a barrage is aimed 
has been taken to be a rectangular one slightly 
smaller than the area covered by a barrage. For exam- 
ple, the rectangular area for short-range barrages is 
120 by 240 ft and that for long-range barrages is 400 
by 400 ft. It is perhaps worthwhile to restate that the 
factors which affect the accuracy and mine clearance 
effectiveness of a rocket barrage are: (1) inherent 
dispersion in range, (2) inherent dispersion in deflec- 
tion, (3) pitch of ship, (4) roll of ship, (5) weaving of 
ship about a straight course, (6) speed of ship, (7) 
radius of clearance, and (8) ranging error of barrage 
as a whole. 

A large number of model barrages were con- 
structed to scale on paper to simulate the actual firing 
of barrages. In constructing each barrage, range 
tables were used for locating the theoretical points 
of impact, and these points were subjected to errors 
or displacements due to factors (1) to (6) above. 
More specifically the displacements due to (3), (4), 
(5), and (6) were applied to the theoretical impact 
points, the characteristics and magnitude of the dis- 
placements having been determined experimentally. 
After imposing displacements due to these factors the 


resulting points of impact of a barrage were then sub- 
jected to random errors (1) and (2), the correct mag- 
nitude and randomness being controlled by tables of 
random numbers. After the final positions of the im- 
pact points had been determined in this manner, 
circles representing clearance were drawn. A large 
number of barrages were constructed for each of three 
sea conditions: No. 1, No. 2, No. 3. Sea conditions 
are reflected in magnitude (amplitude and period) of 
roll, pitch, and weave. The final factor (8) is an 
error which affects the barrage as a whole. It is the 
error caused by misjudging the proper time at which 
to release the barrage as the craft moves in toward 
the beach at full speed. 

In actually simulating an attack on a given rect- 
angular area on the beach, for a given sea state, a 
barrage is taken at random from the set correspond- 
ing to the given sea state, and its general position 
relative to the center of the area is determined by 
using random numbers which simulate the ranging 
error. By averaging the portion of the rectangular 
area (under attack) covered by circles of the barrage 
in a large number of attacks, one obtains an average 
or expected percentage of mines cleared for single 
barrages. Similarly by firing two independently 
aimed barrages at the same area, one can obtain an 
expected percentage of mines cleared by two bar- 
rages and so on for any number of barrages. 

By using this method of analysis, curves giving the 
expected or average percentage of mines not cleared 
in the rectangular area attacked plotted against 
number of barrages aimed at the area, were obtained 
for seven types of mines (Flower Pot, Single Horn, 
Double Horn, Flower Pot with trip wire. Yardstick, 
J93, and TMi43) for the twelve possible combina- 
tions obtainable from three sea states, two rocket 
motor sizes, and two types of explosive. These 
graphs, of which Figure 7 is an example, and the de- 
tails of the analysis have been presented in an AMP 
report.^ 

7 4 CLEARANCE OF PATHS THROUGH 
MINEFIELDS BY AERIAL BOMBING 

A problem in mine clearance in which there was a 
considerable amount of interest on the part of the 
Army Engineer Board, the Joint Army-Navy Ex- 
perimental Testing Board, and the Land Mines 
Committee, NDRC, was that of the feasibility of 
breaching minefields by aerial bombing in circum- 


MINE CLEARANCE BY AERIAL BOMBING 


85 


stances where there would be no hazard to friendly 
troops. Experiments were made under the direction 
of the Army Engineer Board to test this method of 
breaching minefields. The results of these tests were 
rather inconclusive since it did not prove feasible to 


make enough tests to determine the reliability of an 
estimate of the number of bombs required to 
breach a minefield under a given set of conditions. 
Accordingly, AMP was requested to undertake a 
statistical study for the purpose of obtaining reliable 



15 

NUMBER 


20 

OF BARRAGES 


Figure 7. Graph showing average percentage of various types of mines not cleared by WOOFUS barrages. Rockets 
TNT loaded, with 2.25-in. motors. Sea condition— No. 1. Target size— 120 x 240 ft. 


86 


STATISTICAL STUDIES IN MINE CLEARANCE 


estimates of the number of bombs required for clearing 
paths through minefields under various conditions. 

The general method of analysis used in dealing 
with this problem was similar to that employed in 
studying the effectiveness of rocket barrages. In 
other words, model experiments were carried out in 
which aiming errors and other factors affecting the 
location of bombs were simulated by random num- 
bers, and regions of clearance for each bomb were 
idealized as circles. More specifically, for a given 
method of bombing, a given width of minefield, and 
a given radius of clearance a “bombing’’ experiment 
was carried out. A pattern of “bomb” points was 
gradually built up by continued “bombing.” At vari- 
ous times in the experiment, i.e., after certain num- 
bers of “planes” or “formations” had “attacked,” 
an estimate was made as to what percentage of the 
best path (chosen by inspection) was cleared of 
mines. As an experiment progresses this percentage, 
of course, increases. The experiment was terminated 
after 40 to 80 “planes” had been “flown” (depending 
on the bombing conditions). Repeating this experi- 
ment a number of times yielded enough information 
to enable one to construct a curve showing the aver- 
age or median percentage of clearance (called the 50 
per cent curve) of the best path plotted against the 
number of “planes attacking.” Similar curves based 
on any percentile (rather than median) percentage 
of clearance of the best path could also be drawn. In 
fact, the 90 percentile percentage curve (called the 
90 per cent curve) is useful, since it gives a figure, for 
a given number of planes, representing the percent- 
age of clearance which would be exceeded 90 per cent 
of the time in a large number of similar repeated 
attacks. 

The details of this method of analysis have been 
presented in an AMP report.^ The method has been 
applied to five types of bombing, together with their 
associated bombing errors. They are: 

1. Dive bombing with small aiming error. 

2. Dive bombing with large aiming error. 

3. Medium altitude bombing with single medium 

bombers. 

4. Medium altitude bombing with single heavy 

bombers. 

5. Formation bombing with heavy bombers. 

The widths of minefields considered were 300, 600, 
and 900 ft. The radii of clearance used were 15, 20, 
25, 30, and 40 ft. The path selected under each set 
of conditions as best is the one that would be most 
nearly covered by circles of the given radii. 


The result of these statistical experiments were 
presented in thirty graphs.^ The thirty graphs corre- 
spond to the thirty possible combinations obtainable 
from the five bombing methods, the three minefield 
widths, and the two levels of confidence (50 per cent 
curves and 90 per cent curves) described earlier. On 
each graph are five curves corresponding to the five 
different radii of clearance considered. Each curve 
shows percentage clearance of mines in best path 
plotted against number of planes attacking, for the 
conditions specifying the chart and the particular 
curve in the chart. Figure 13 of Chapter 4 shows one 
of these thirty graphs. 

The principal result of this study was the conclu- 
sion that a large number of bombs would be required 
to breach minefields satisfactorily, except possibly in 
the case of dive bombing against narrow minefields 
with AN Mk47 depth bombs equipped with air burst 
fuzes (40 ft radius of clearance). 

An earlier study^ of the possibility of clearing paths 
through minefields by aerial bombing, under rather 
more restricted conditions, was carried out by AMP 
upon request of the Committee on the Demolition 
of Landing Obstacles. In this study two possibilities 
of clearance of path through a minefield were con- 
sidered: (1) the release of a sufficient number of 
bombs to clear a path at a predetermined location 
across the minefield of width B, and (2) the release 
of several trains of sufficiently closely spaced bombs 
joined in a zigzag fashion so as to clear a zigzag path 
through the minefield. 

In the case of possibility (1), the path across the 
minefield was laid out in advance. This is to be con- 
trasted with the approach to the problem of land 
mine clearance taken in the study'" described earlier, 
in which the path was chosen after the bombing had 
been carried out, thus diminishing to some extent the 
number of bombs required. The types of bombing 
considered were formations of nine and eighteen 
planes, each plane carrying a specified number n of 
bombs to be dropped in train. The problem consid- 
ered was that of detonating a mine at a designated 
place in the path by a specified number F of forma- 
tions, where the planes in each formation were as- 
sumed to have dropped their bombs in formation 
from an altitude of 8,000 to 10,000 ft aiming at the 
center of the path. If the specified mine is located in 
a corner of the path, then P is smaller than the prob- 
ability of detonation for a mine location anywhere 
else in the path. Thus, lOOP represents the upper 
bound of the expected percentage of land mines 


MINE CLEARANCE BY AERIAL BOMBING 


87 


within the path which will be left uiiexploded by at- 
tacks from F formations. Thus, if P is very small it 
may be presumed that practically all mines within 
the path will be cleared. A nomogram was con- 
structed which provided a relationship between 
B, R (radius of detonation), F, n, and P, so that 
any one of the variables could be determined for 


specified values of the remaining four variables. 

In considering possibility (2), i.e., the problem of 
establishing a path by joining tight strings (or trains) 
of bombs in a zigzag fashion, it was found that aim- 
ing errors- and dispersion of bombs in train had to 
be of magnitudes too small to be realistic under com- 
bat conditions. 


Chapter 8 

STATISTICAL ANALYSIS OF THE PERFORMANCE 
OF HEAT-HOMING DEVICES 


81 THE PROBLEM 

I N 1944 an aerial experiment was carried out by 
the Optics Section, Bureau of Ordnance [BuOrd] 
to survey the thermal characteristics of various types 
of targets, and to determine the effectiveness of sev- 
eral newly developed heat-homing devices [HHD] 
in detecting these targets. The targets included fac- 
tories and docks in various parts of the country and 
ships off shore. In carrying out these experiments the 
question arose as to what observations should be re- 
corded and how they should be analyzed in order to 
determine the reliability with which a given device 
would indicate the presence and direction of the 
target under various conditions. AMP was requested 
to assist with the statistical aspects of this problem. 

82 the experimental set-up 

In setting up the experiment, provision was made 
for testing three types of HHD’s namely Type A, 
Type B, and Type C.^ Type A was designed for 
measuring the thermal intensity of heat signals, while 
Type B and Type C were designed to indicate direc- 
tion of thermal centers. Type B was gradually im- 
proved and used as the homing device for the FELIX 
(VB-6) heat-homing, high-angle bomb. These three 
experimental devices were mounted in an airplane in 
conjunction with a Farrand instrument and a 16-mm 
camera. The Farrand instrument was used to meas- 
ure the heat intensity of a target and to indicate the 
thermal center of the target. The intensity was re- 
corded on a waxed tape. The Farrand instrument 
and the camera (camera No. 1) were locked together 
so as to “see” the same picture on the ground. They 
could be rotated about an axis parallel to the line 
of flight, and hence could be tracked across the target 
simultaneously as the plane moved over the target. 
The three candidate instruments were free to be 
tracked “above” or “below” the target as seen by 
an observer oriented in the plane so as to be facing 
the right side of the plane and looking down. In this 
orientation, the left of the field of view was therefore 
the forward direction of flight. An instrument panel 

^ Type A refers to the Aiken HHD, Type B to the Bemis 
HHD, and Type C to the Offner HHD. 


was designed for recording the signals yielded by the 
three candidate instruments. For Type B and Type C 
there were, in each case, four lights arranged as shown 
in Figure 1. When either of these two instruments 
performed perfectly, then when it was aimed at a 
point up and to the left of the target (high thermal 
center) with respect to the oriented observer men- 

UP 

4 • I 

LEFT* •RIGHT 

3 • Z 

DOWN 

DIRECTION 

OF FLIGHT 

Figure 1. Diagram showing arrangement of lights for 

Type B and Type C heat-homing devices. 

tioned above, the two lights marked down and right 
(or more briefly the pair 2) would flash on, thus 
indicating that the target was down and to the right. 
Similarly, this holds for other aimed positions with 
respect to the target. This instrument panel was 
photographed by a second camera (No. 2) which was 
synchronized with the camera aimed at the target. 

Thus, as the plane flew over the target, camera 
No. 1 and the Farrand instrument would be aimed 
at the ground so as to track across the target from 
left to right. The Type A, Type B, and Type C in- 
struments could be aimed “above” or “below” or 
‘‘on” the target, independently of each other and of 
the Farrand and camera No. 1 set-up. Camera No. 2 
photographed the instrument panel which showed 
the signals of the three candidate instruments. The 
operators of these three instruments signified whether 
on a given run they were tracking above or below 
the target, and by how much (in mils). 

8 8 ANALYSIS OF THE DATA 

The data on a given run across a given target 
was compiled from information on the two films 
taken by synchronized cameras No. 1 and No. 2, and 


88 




ANALYSIS OF THE DATA 


89 



Figure 2. Photograph showing the direction of target indications in four runs across the Emerson Turret Plant. Runs 
9005 (B) and 9006 (A) were traced below and above the target respectively, while runs 9004 and 9007 were traced on the target. 


from a data sheet indicating whether the Type B 
(Type A or Type C) instrument was being tracked 
‘‘above/’ “below,” or “on” the target. A double pro- 
jector was devised for running the two films simul- 
taneously. A coordinate system was set up on the 
screen on which the target film was projected in 
such a way that each dimension of a single projected 
frame of the film was divided into five equal parts 


(screen units) — the area of the frame therefore being 
divided into twenty-five equal rectangles. Thus, the 
position of the target could always be specified on 
each frame. Therefore, by running the films through 
the projector, one is able to reconstruct, for a given 
airplane run, the direction of thermal center indi- 
cated by Type B or Type C as it is being tracked 
“above,” “below,” or “on” the target. Figure 2 shows 







90 


PERFORMANCE OF HEAT HOMING DEVICES 


an example of how these indicated directions can be 
reconstructed along a Type B or Type C track. The 
four tracks represent Type C runs corresponding to 
four airplane runs across the target North-South, 
South-North, East-West and West-East. The short 
spurs represent the directions of thermal centers in- 
dicated by Type C at various points of the path. The 
letters A and B after the numbers indicate Type C 
tracking above and below the target, respectively. 
The numbers are used to identify the runs. A num- 
ber without A OY B means tracking on the target. 

In addition to being able to reconstruct a track for 
Type B or Type C with indicated directions of target 
(thermal centers) along the track, it was possible to 
determine the reliability with which either device 
would indicate direction of thermal centers of a given 
intensity as measured by the Farrand. In other 


Table 1. Type C HHD target signals for a series of 
airplane runs. 


Track 

Up 

Target position 
Down Right 

Left 

On target 

Right 

1,604 

999 

1,023 

1,580 

Left 

1,600 

801 

764 

1,637 

Total 

3,204 

1,800 

1,787 

3,217 

Above target 

Right 

123 

104 

79 

148 

Left 

171 

123 

74 

220 

Total 

' 294 

227 

153 

368 

Below target 

Right 

251 

81 

166 

166 

Left 

180 

64 

148 

96 

Total 

431 

145 

314 

262 


words, one could make up a table showing frequency 
(number of frames) of indicated directions of thermal 
centers versus actual direction. Table 1 is an example 
of such a table for a series of airplane runs made in 
March 1944, using a model of the Type C device. It 
will be noted that Type C was tracked below the 
target in the case of a total of 576 frames and there 
are 431 “up” signals (correctly indicated signals). 
Similarly, this holds for other directions. The reli- 
ability of calling signals depends not only on the in- 
tensity of heat at the main target, but also on the 
presence of other thermal centers and on the sensi- 
tivity of the instrument. As may be expected, some 
targets were found to be thermally indistinguishable 
from the background, or even colder than back- 
ground (for example ships early in the morning). 

Reliabilities of direction signals of the Type B and 


Type C instruments were statistically determined for 
a wide variety of targets, altitudes, times of day, and 
flight into or away from the sun. The relation be- 
tween expected signal amplitude and percentage of 
correct signals as a function of distance from center 
was investigated. The behavior of signals for the 
thermal distribution found in land and water back- 
grounds was also thoroughly analyzed. The average 
duration of sustained signals from these two devices 
was determined under various conditions. The re- 
producibility of signal patterns for repeated airplane 
runs was studied. The results of these studies were 
reported in detail in several AMP reports. 

The main conclusions of the work reported in these 
reports, which pertain primarily to Type C, may be 
summarized as follows. 

1. There was a distinct sun glare effect on all runs 
on which the HHD detector was focused in the direc- 
tion of the sun. 

2. There was a noticeable signal lag in indicating 
the target as the HHD swept over the target. This 
is an inherent feature of any electrical system. 

3. The instrument clearly indicated the thermal 
differences between land and water. This difference 
was so great that it masked any thermal differences 
between the land and objects built on the land or 
ships anchored near the shore. 

4. The results over ships at sea were much poorer 
than over land targets. 

5. Almost perfect performance was obtained over 
very hot targets such as oil fire, and blast furnaces. 

6. The results were definitely poorer at altitudes 
above 6,000 ft than below this altitude. Also poor 
results were obtained with heavy overcast skies. 

7. Cold-homing might be feasible up to a few min- 
utes after sunrise. Poor results could be expected for 
at least an hour after sunrise. 

8. The target indications improved as the center of 
the HHD approached the heat center. The improve- 
ment closely paralleled the theoretical relative signal 
amplitude as a function of distance from the heat 
center. 

84 STUDY OF IMPROVED TYPE B HHD 

During the latter part of 1944 an improved model 
of the Type B HHD was constructed by Division 5, 
NDRC (the original designers of the Type B instru- 
ment), and tested early in 1945. AMP was requested 
to assist in the statistical analysis of data obtained 
in this experiment. 


STUDY OF IMPROVED TYPE B HHD 


91 


The performance of this improved device was 
evaluated in terms of the reliability of its production 
of direction signals in a 10° circle about the target 
when used at an altitude of 10,000 ft. The perform- 
ance was found to be good for most of the targets 
considered. The experimental procedure used in the 
Division 5 tests was improved over that used in the 
earlier Bureau of Ordnance tests and the statistical 
analysis was accordingly modified. The essential 
changes in the new Type B tests were: 

1. The FELIX eye and the camera which photo- 
graphed the target area were synchronized and had 
the same field of view. On the BuOrd surveys a 
different person operated the camera and each HHD ; 
hence, it was not possible to guarantee that the pho- 
tographed area was actually the area at which a 
given HHD was directed. 

2. Signal lights, which gave the HHD target indi- 
cations, were photographed on the same film as the 
target area on the FELIX survey, while separate 
films were used on the BuOrd surveys. 

3. The HHD and camera were moved only up and 
down on the BuOrd flights and were locked in 
azimuth ; hence, only one crossover of the target area 
was obtained on a given run over the target. On the 
FELIX flights the operator repeatedly swept the 
camera (locked with FELIX) across the target both 
up-down and right-left on a given run. As a result of 
this latter innovation, much more information was 
collected on the relative stability of the HHD in 
indicating the target center. Also accurate data were 
then available on the actual lag in signaling. In or- 
der to utilize these data, the points of signal crossover 
(changing of lights from up to down, or right to left, 
or vice versa) were determined. The average center, 
the average variability of this center location, and 
the signal lag were easily determined for every target 
area surveyed. 

In this experiment it was possible to estimate that 
the effect of the small signal delay (due to an elec- 
trical lag of 0.16 sec in the system) of the Type B 
HHD was about 3 frames of a 16-mm film. This 
created a corresponding small bias in the HHD’s 
estimate of thermal center in the direction of flight 
over the target. Figure 3 shows graphically the effect 
of this signal lag. This figure represents the results 
of a laboratory calibration run in which the HHD was 
placed 15 ft above the heat source and was swept 
across the heat source from this fixed position. The 
coordinate system is in terms of screen units. 

The details of the statistical analysis of the per- 


formance of the improved Type B HHD have been 
reported in an AMP report.® The principal results 
of the analysis may be summarized as follows. 

1. Since the maximum area seen by the FELIX 
^‘eye” is approximately a circle on the ground of a 
10° radius (e.g., at a 10,000-ft altitude, this radius 


x*o 



Figure 3. Target indications^by FELIX when 
moved about a heat source. 


The plotted path represents the course of the center of the field 
of the camera (and the Type B, which was locked to the camera). 

The arrows at the individual points of the path show the direction 
in which FELIX indicated the heat source. If there were no signal 
lags, the arrows should point towards the unshaded circle. The num- 
bers at the points of the path are frame numbers when the film 
runs at the rate of 16 frames per sec. 

would be about 1,750 ft), the success of the survey 
unit over a given target area was evaluated in terms 
of its operation within a 10° circle centered at the 
target center. For nine of the targets, the HHD was 
swept over the target areas in such a manner that the 
center of vision covered approximately a circular area 
of at least a 10° radius. Good results were obtained 
over five of the targets for a full 10° circle, over one 
target for a full 10° circle on six out of eight runs, 
and over two targets for a circle of somewhat less 
than the full 10° radius. Definitely bad results were 
obtained over one of these nine target areas. Over the 
other four target areas, the center of the HHD was 
swept over an area of less than a 10° radius. Correct 
target indications were obtained over all the area 
scanned for three of these targets, while good indica- 
tions were given for a circle of only a 5° radius over 
the fourth target. 

2. Overcast skies and snow on the ground tended 
to reduce the differential thermal effects between the 
target area and such areas as parking lots, cleared 
spaces, hillsides, and water. 


92 


PERFORMANCE OF HEAT-HOMING DEVICES 


3. Thermal differences between the target area and 
an adjacent water area were reversed at night unless 
the plant had much internal heat. 

4. The variability of the HHD operation over a 
given target tended to be smaller if the background 
were relatively uniform, if the target consisted of one 
main building instead of several buildings, and if 
there were a large thermal differential between the 
target and the surrounding area. Also, of course, the 
signals are more consistent when the instrument is 
centered over a point near the thermal center of 
target. 

5. There was insufficient data to make possible any 
general statements on the minimum thermal in- 
tensity necessary for successful HHD operation. 
However, successful daytime operation was obtained 
over one plant with a thermal intensity as low as 1.5 
ergs per sq cm per sec. 

6. The estimate of the center of a target area on a 
given run was generally slightly biased in the direc- 
tion of flight over the target. 

85 FURTHER POSSIBLE STATISTICAL 
STUDIES OF THE PERFORMANCE OF HHD’S 

The experience of the AMP in the analysis of data 
regarding the performance of HHD’s indicates that 
if further tests are carried out the following points, 
at least, should be considered. 


1. Survey data are needed over such intense targets 
as steel mills and blast furnaces. 

2. To further study the effect of fog and overcast 
sky, etc., more runs should be made over the same 
plant under varying weather conditions. 

3. In tracking about a target, the instrument 
should be tracked counterclockwise part of the time. 
It was always tracked clockwise in the present 
surveys. 

4. More information is needed on the effect of 
water near a factory area, especially as a function 
of time of day and sunlight. 

5. The analysis of target areas could be much 
better applied to other targets if the radiation in- 
tensity were mapped over the region of the target. 

Of course, final classification of targets with 
regard to their susceptibility to heat-homing de- 
vices must await actual bombing tests. In order to 
make such tests it seems necessary that radia- 
tion intensity maps of typical factory areas must 
be made and reproduced approximately for the 
bombing tests. 

Also, the determination of the best lag must be 
done by means of experiments with actual bombs, 
since the adjustment of the lag in the bomb depends 
on the aerodynamic constants of the bomb itself 
(damping ratio and natural frequency). The lag of 
the survey unit was made small in the interests of 
convenience in film reading. 




BIBLIOGRAPHY 


Numbers such as AMP-1 1-M3 indicate that the document listed has been microfilmed and that its title appears m the 
microfilm index printed in a separate volume. For access to the index volume and to the microfilm, consult the Army 
or Navy agency listed on the reverse of the half-title page. The following abbreviations are used: 


BuOrd Bureau of Ordnance 

SRG-P Statistical Research Group — Princeton 

BRG-C Bombing Research Group — Columbia 
AMG-C Applied Mathematics Group — Columbia 

Summary 

1. “Sequential Tests of Statistical Hypothesis,” Abraham 
Wald, The Annals of Mathematical Statistics, Vol. XVI, 
No. 2, June 1945, pp. 117-186. 

2. Sequential Analysis of Statistical Data: Applications, H. A. 

Freeman and Abraham Wald, prepared for AMP, 
Columbia University Press, Revised AMP Report 30.2R 
[OEMsr-618], SRG-C, Sept. 15, 1945. AMP-21. 1-M4 

Chapter 2 

^ 1. Miscellaneous Probability Tables, H. H. Germond, AMP 
Note 14, July 1944. AMP-1 1-M3 

ir 2. Aliscellaneous Probability and Statistical Tables and Graphs, 
OEMsr-860, SRG-P 156, AMP Note 24, October 1945. 

AMP-1 1-M6 

^ 3. Scatter Bombing of a Circular Target, H. H. Germond and 

Cecil Hastings, Jr., OSRD 4572, OEMsr-818 and OEMsr- 
1007, BRG-C 120, AMG-C 302, AMP Report 10.2R, 
May 1944. AMP-803.4-M2 

4. Some Quantitative Information which Bears on the General 

Problem of Bombing Systems. AMP Memorandum 11.13M, 
Sept. 30, 1943. AMP-806-M2 

5. Note on Determination of Aiming Points and Number of 
Bombs for Bombing Operations, OSRD 5317, OEMsr-860, 
SRG-P 138, AMP Note 17, June 1945. AMP 806-M3 

6. Analytical and Statistical Studies of Certain Guided 
^ Missiles, SRG-P 158, AMP Report 112.2R, October 1945, 

jjp. 44-50. AMP-806-M4 

7. Air-to-Air Bombing, Memorandum No. 2 to Steering Com- 
mittee for AC-92 from Sub-Committee on Bombing Effective- 
ness, OEMsr-860, SRG-P 61, Sept. 23, 1944. 

AMP-803.5-M2 

8. Air-to-Air Bombing Probabilities for Three Types of Fuze, 

OEMsr-1365, SRG-P 69, Oct. 26, 1944. AMP-704-M11 

Chapter 3 

1. Preliminary Report on a Study of Train Bombing, OSRD 
1869, OEMsr-65, Service Project AC-27, Report to 
Services 33, Section D-2, Fire Control, NDRC, AMP 
Report ll.lR, Aug. 25, 1942. AMP-803. 1-Ml 

^ 2. The Probabilities of Hitting, in Train Bombing, Rectangular 
Targets of Proportion One-by-Six or One-by-Nine, OSRD 
1278, Report to Services 53, Division 7, NDRC, AMP 
Report 11.3R, Mar. 10, 1943. AMP-803. 1-M4 

3. The Theory of Midtiple Hits on Multiple Targets in Train 

Bombing, Appendices A and B, Jerzy Neyman, OSRD 
1476, Report to Services 55, Division 7, NDRC, AMP 
Report 11. 4R, May 10, 1943. AMP-803. 1-M5 

4. Tables of Probabilities of at Least One, Two, Three, Four, 

and Five Hits on Rectangular Targets in Train Bombing, 
OSRD 3882, OEMsr-818, BRG-C 108, AMP Report 
ll.lOR, June 1944. AMP-803. 1-M9 

5. Probabilities of at Least One, Two, Three, Four, and Five 


AMP Applied Mathetics Panel 

SRG-C Statistical Research Group — Columbia 

AMG-P Applied Mathematics Group — Princeton 


Hits on Rectangular Targets in Train Bombing when the 
Dispersion is Equal to the Target Width, OSRD 4644, 
OEMsr-818, BRG-C 122, AMP Report 11.13R, Decem- 
ber 1944. AMP-801. 1-M5 

6. The Character of the Train Bombing Probability Curve at 

the Point for Zero Spacing, J. D. Williams, OEMsr-818, 
BRG-C 112, June 16, 1944. AMP-803. 1-M 10 

7. Note on the Direction of Attack with a Train of N Bombs 
when the Aiming Errors in Range and Deflection are 
Unequal, Jan Schilt, OEMsr-818, BRG-C 116, July 1944. 

AMP-803. 1-M 11 

8. Train Bombing, Outline of Pnncipal Results of Statistical 
Studies Conducted by the Applied Mathematics Panel, 
OSRD 1784, AMP Report 11.5R, Aug. 12, 1943. 

AMP-803. 1-M8 

9. An Empirical Verification of Tables on Multiple Hits, 

Mark W. Eudey, OSRD 1810, OEMsr-817, AMP Report 
11. 7R, BRG-Statistical Laboratory, University of Cali- 
fornia, Aug. 10, 1943. AMP-801. 2-M2 

10. Area Bombing Probabilities, H. H. Germond, OSRD 4321, 

OEMsr-1007 and OEMsr-818, BRG-C 119, AMG-C 237, 
AMP Report 11.12R, July 1944. AMP-803.2-M4 

11. Note on Mission Planning Against an Important Target, 
Jan Schilt, BRG-C 81, November 1943. AMP-801. 1-M12 

12. Note on Mission Planning Against Multiple Targets, 
Jan Schilt, OEMsr-818, BRG-C 82, November 1943. 

AMP-801. 2M-4 

13. Note on Spacing for Group Attacks and Alultiple Hits, 
Jan Schilt, OEMsr-818, BRG-C 83, November 1943. 

AMP-801.2-M5 

14. Observations Concerning Multiple Attack and Multiple Hit 

Theory, Jan Schilt, AMP Memorandum 11.12M, Columbia 
University, September 1943. AMP-801. 2-M3 

15. Design of a Bomb Spacing Calculator, H. H. Germond, 

OEMsr-818, (Rewrite of BRG 77), BRG 100, Apr. 26, 
1944. AMP-80 1.3-M2 

16. Simplified Rule for Determinmg Spacing in Train Bomb- 
ing on Stationary Targets, Emma Lehmer, BRG 87, 
Statistical Laboratory, University of California, Decem- 
ber 1943. 

17. A Study of the Seriousness of the Effects, in the Planning 

and Executing of Bombing Missions, of Mis-Estimates 
of the Standard Errors of Aiming and Dispersion, OSRD 
1149, Report to Services 46, Division 7, NDRC, AMP 
Report 11.2R, Jan. 12, 1943. AMP-803.1-M3 

18. On the Effects of Failing to Make a Correction of Trail in 

Train Bombing so as to Place the Center of Train on the 
Center of Target, Evelyn Fix, OSRD 1800, AMP Report 
11. 6R, BRG-Statistical Laboratory, University of Cali- 
fornia, August 1943. AMP-803. 1-M7 

19. Methods of Estimating Standard Errors of Aiming from 

Operational Data, Jerzy Neyman, OSRD 3342, OEMsr- 
817, AMP Report 11. 9R, BRG-Statistical Laboratory, 
University of California, January 1944. AMP-801. 1-M3 




93 


94 


BIBLIOGRAPHY 


20. High Level and Medium Level Bridge Bombing, R. I. Wolff, 

(Revised paper of BRG 105), BRG-C 115, AMP Memo- 
randum 11.14M, July 1944. AMP-804.4-M2 

21. The Problem of Optimum Spacing in Low Altitude APQ-5 

Train Bombing, OEMsr-860, SRG-P 141, AMP Memo- 
randum 11.15AI, July 1945. AMP-803. 3-M7 

22. Clearance of Paths Through Minefields by Aerial Bombing, 

OSRD 6082, OEMsr-860, SRG-P 151, AMP Report 
178.2R, August 1945. AMP-902-M4 

23. Proof that the Optimum Spacing for Smking Probabilities 
is Independent of the Number of Attacks, J. D. Williams, 
OEMsr-818, BRG-C 109, Feb. 6, 1944. AMP-804.2-M4 

24. Sinking Probabilities and Coefficients for at Least One Hit, 
at Least Two, etc., Jan Schilt, BRG-C 89, Dec. 27, 1943. 

AMP-804.2-M5 

25. Preliminary Report on Scatter Bombing, H. H. Germond 

and Cecil Hastings, Jr., OSRD 904, Report to Services 34, 
Section D-2, Fire Control, NDRC, AMP Report 10. IR, 
September 1942. AMP-803. 4-Ml 

26. Scatter Bombing of a Circular Target, H. H. Germond and 

Cecil Hastings, Jr., OSRD 4572, OEMsr-818 and OEMsr- 
1007, BRG-C 120, AMG-C 302, AMP Report 10.2R, 
May 1944. AMP-803.4-M2 

27. The Probability of at Least One Hit and the Average Number 

of Hits for Salvos with Various Dispersions Against 
Rectangular Targets, OSRD 4804, OEMsr-818, BRG-C 
123, AMP Report 10.3R, Jan. 1945. AMP-801. 1-M6 

28. Air-to-Air Bombing, Memorandum No. 2 to Steering Com- 

mittee for AC-92 From Sub-Committee on Bombing Effec- 
tiveness, SRG-P 61, Sept. 23, 1944. AMP-803.5-M2 

29. Analytical and Statistical Studies of Certain Guided 

Missiles, OEMsr-860, SRG-P 158, AMP Report 112.2R, 
October 1945, pp. 19-22. AMP-806-M4 

Chapter 4 

1. Miscellaneous Probability and Statistical Tables and 
Graphs, OEMsr-860, SRG-P 156, AMP Note 24, Oct. 1945. 

AMP-1 1-M6 

2. Anahjtical and Statistical Studies of Certain Guided 

Missiles, OEMsr-860, SRG-P 158, AMP Report 112.2R, 
October 1945. AMP-806-M4 

2a. Ibid., pp. 50-63. 

3. Average Proportion of Hits on a Rectangular Target when 
Bombed by a Uniform Rectangular Pattern, OEMsr-860 
and OEMsr-618, SRG-P 96 and SRG-C 426, Jan. 22, 1945. 

AMP-801. 1-M7 

4. Cooperative Study on Area Bombing, Mark W. Eudey, 

Evelyn Fix, E. Lehman, D. H. Lehmer, Emma Lehmer, 
Jerzy Neyman, J. Robinson, M. Shane, and E. L. Scott, 
OEMsr-817, (Report submitted to AMP, NDRC), 

Statistical Laboratory, University of California, Oct. 1944. 

AMP-803.2-M3 

5. Distribution of the Percentage of Hits when Uniform Square 

Bomb Patterns are Dropped on Rectangular Targets, 
OSRD 5212, OEMsr-860, SRG-P 119, AMP Report 
184.1R, April 1945. AMP-803.2-M6 

6. A Coverage Problem Associated with Bombing, OEMsr-860 
SRG-P 157, AMP Report 173. IM, October 1945. 

AMP-803.2-M8 

7. Cooperative Study on Probability of Exploding Land Mines 

by Bombing, Mark W. Eudey, Evelyn Fix, Emma Lehmer, 
Jerzy Neyman, and E. L. Scott, OEMsr-817, (Summary 
Report submitted to Division 2 and AMP, NDRC), 
Statistical Laboratory, University of California, April 1, 
1944. AMP-902-M2 

8. Dependence of the Percentage of Hits on Pattern Area and 
Mean Radial Error, Derived from Operational Data on 


Formation Bombing, OSRD 5416, OEMsr-860, SRG-P 130, 
AMP Report 174. IR, July 1945. AMP-803.2-M7 

9. An Empirical Determination of the Dependence of Pattern 
Area and Mean Radial Aiming Error on Certain Operating 
Factors in Formation Boinbing, OEMsr-860, SRG-P 159, 
AMP Report 174.2R, October 1945. AMP-803.2-M9 

10. Clearance of Paths Through Minefields by Aerial Bombing, 
OSRD 6082, OEMsr-860, SRG-P 151, AMP Report 
178.2R, August 1945. AMP-902-M4 

'^11. Attacks on a Maneuvering Target by a Small Number of 
Level Bombers, J. D. Williams, OSRD 1905, AMP 
Report 11. 8R, Columbia University, September 1943. 

AMP-804.4-M1 

12. A Photo-Electric Instrument for Rapid Bomb-Fall Analysis 
and Damage Prediction, D. H. Lehmer, OSRD 5641, 
OEMsr-817, AMP Report 190. IR, University of Cali- 
fornia, May 1945. AMP-804. 4-M4 

Chapter 5 

1. Area Bombing Probabilities, H. H. Germond, OSRD 4321, 

OEMsr-1007 and OEMsr-818, BRG-C 119, AMG-C 237, 
AMP Report 11.12R, July 1944. AMP-803.2-M4 

2. Miscellaneous Probability and Statistical Tables and 

Graphs, OEMsr-860, SRG-P 156, AMP Note 24, October 
1945, pp. 1-19. AMP-1 1-M6 

3. Scatter Bombing of Bomber Strips, J. D. Williams, BRG-C 

106, May 28, 1944. AMP-803.4-M3 

4. A Study of the Effectiveness of IB Aerial Attacks on German 
Industrial Targets, [Part] I. Probability of M47 Starting 
a Serious Fire in an Industrial Building, E. W. Barankin, 
E. R. Dempster, and others, OEMsr-817, AMP Memo- 
randum 190. IM, University of California, September 1945. 

AMP-804.3-M2 

5. A Study of the Effectiveness of IB Aerial Attacks on German 

Industrial Targets, [Part] II. Analysis of RAF Area 
Attack of 22123 October 1943, on Kassel, with Special 
Reference to Effectiveness of the M50 Magnesium Bomb in 
Setting Fire to Industrial Buildings, Evelyn Fix, OEMsr- 
817, AMP Memorandum 190. 2M, University of Cali- 
fornia, Seiitember 1945. AMP-804. 3-M3 

6. A Study of the Effectiveness of IB Aerial Attacks on German 
Industrial Targets, [Part] III. Summary Report on Fire 
Raising Effectiveness of Incendiary Bombs M47 and M50 
Used Against German Industrial Targets, Leo A. Aroian, 

E. W. Barankin, and others, OPlMsr-817, AMP Memo- 
randum 190.3M, LTniversity of California, September 1945. 

AMP-804.3-M4 

7. A Ne w Vulnerability Equation in the Analysis of Incendiary 

Raids, J. Bronowski and Jerzy Neyman, OSRD WA- 
3924-14, REN Report 471, Research and Experiment 
Dept., Ministry of Home Security, Great Britain, Dec. 1, 
1944. AMP-804. 1-Ml 

8. Effect of Cluster Spacing on IB Densities in Area Attacks, 

F. Garwood and J. Bronowski, OSRD II-5-5777(S), 

REN Report 372, Research and Experiment Dept., 

Ministry of Home Security, Great Britain, April 28, 1944. 

AMP-803.2-M1 

9. Bomb Damage, Mary L. Shane, Progress Report 2, 

Feb. 16, 1945. AMP-804.4-M3 

[A Stochastic Model of Incendiary Raids] Jerzy Neyman, 
Progress Report 3, Feb. 28, 1945. AMP-803. 2-M5 

[Distribution of Bombs] Progress Report 4, Mar. 9, 1945; 

AMP-801. 1-M 8 

[Distribution of Bombs] Jerzy Neyman, Progress Report 
5, Mar. 14, 1945. AMP-801. 1-M9 

Estimates of the Frequency Constants of the Distribution of 
Bombs, Jerzy Neyman, OEMsr-817, Service Project AN- 


BIBLIOGRAPHY 


95 


23, Progress Report 6, (Submitted to the Joint Target 
Group AC/AS Intelligency Hdqs. AAF), Statistical 
Laborator}', University of California, Mar. 20, 1945. 

AMP-SOl.l-MlO 

10. The Effectiveness of Large Blast Bombs Against German 
Housing, Jerzy Neyman, OEMsr-817, AMP Memorandum 
190.4M, University of California, September 1945. 

AMP-804.3-M1 

Chapter 6 

1. Optimum Angle off the Bow for Torpedo Attacks, OEMsr- 

618, SRG-C 102, AjMP Memorandum 71. 2M, Nov. 1, 
1943. AMP-405.2-M1 

2. Optimum Spread Angles for Destroyer Torpedo Salvos, 

OSRD 3939, OEMsr-618, SRG-C 266, AMP Report 
71. IR, July 1944. AMP-405.2-M2 

3. Torpedo Training Exercises, Serial 1074, issued by 
Commander Destroyers, Pacific Fleet, IMar. 2, 1943. 

4. Efficient Paths for a Zig-Zag Torpedo, OEMsr-618, SRG-C 
338, AMP Report 105. IR, October 1944. AMP-405. 5-M2 

5. Lead Angles for Aerial Torpedo Attacks Against Turning 

Ships, OSRD 3867, OEMsr-618, SRG-C 190, AMP 
Report 8.1R, July 1944. AMP-405.1-M6 

6. Tables of Aircraft Torpedo Lead Angles, OSRD 5097, 
OEMsr-618, SRG-C 453, AMP Report 8.2R, May 1945. 

AMP-405. 1-M9 

Chapter 7 

1. Expected Clearance of German and Japanese Antitank and 
Antipersonnel Mines by Explosive Mine Clearing Devices, 
OEMsr-860, SRG-P 136, AMP Report 178. IR, June 1945. 

AMP-902-M3 

2. Curves Showing Expected Percent of Tanks which Will 
Cross Minefields ivithout Sinking Mines, OEMsr-860, 
SRG-P 121a, AMP Memorandum 178. IM, April 1945. 

AMP-901.2-M1 

3. Expected Percent of Tanks Passing Through Minefields 


without Striking Mines, OEMsr-860, SRG-P 122, AMP 
Memorandum 178.2M, May 1945. AMP-901. 2-M2 

4. Clearance of Mines by Rocket Barrages from the Woof us, 
OSRD 5640, SRG-P 150, AMP Report 161. IR, Aug. 1945. 

AMP-902-M5 

5. Clearance of Paths Through Minefields by Aerial Bombing, 
OSRD 6082, SRG-P 151, AMP Report 178.2R, Aug. 1945. 

AMP-902-M4 

6. Cooperative Study on Probability of Exploding Land Mines 

by Bombing, Mark W. Eudey, Evelyn Fix, Emma Lehmer, 
Jerzy Neyman, and E. L. Scott, OEMsr-817, (Summary 
Report submitted to Division 2 and AMP, NDRC), 
Statistical Laboratory, University of California, Apr. 1, 
1944. AMP-902-M2 

Chapter 8 

1. Analysis of Operation of Heat-Homing Devices during 
March, 1944, First AMP Report on BuOrd Flight Project 
70, SRG-P 38, AMP Report 127. IR, June 1944. 

AMP-805-M1 

2. Analysis of Operation of Heat-Homing Devices near 

St. Louis and Nashville on April 25, 1944, Second AMP 
Report on BuOrd Flight Project 70, SRG-P 47, AMP 
Report 127. 2R, August 1944. AMP-805-M2 

3. A nalysis of Operation of Heat-Homing Devices during Alay, 

1944, BuOrd Flight Project 70, Report 3, SRG-P 59, 
AMP Report 127. 3R, September 1944. AMP-805-M3 

4. Analysis of Operation of Heat-Hoining Devices May 31 to 
June 24, 1944, BuOrd Flight Project 70, Report 4, 
SRG-P 71, AMP Report 127.4R, November 1944. 

AMP-805-M4 

5. Analysis of Operation of Heat-Homing Devices after April, 

1944, BuOrd Flight Project 70, Report 5, SRG-P 92, 
AMP Report 127. 5R, January 1945. AMP-805-M5 

6. Analysis of Operation of Felix Heat-Homing Device over 

Targets Near Boston, Massachusetts, during October 1944 
and January 1945, OSRD 5006, OEMsr-860, SRG-P 120, 
AMP Report 112.1R, April 1945. AMP-805-M6 


OSRD APPOINTEES 
APPLIED MATHEMATICS PANEL 


Chief 

Warren Weaver 


Deputy Chief 

Thorton C. Fry 

Acting Chief, May 29, 1945 to April 5, 1946 


Technical Aiders 

B. H. Colvin 
II. H. Germond 
Cecil Hastings, Jr. 

Myrtle R. Kellington 
Margaret S. Piedem 


Mina S. Rees 

I. S. SOKOLNIKOFF 
D. C. Spencer 

S. S. Wilks 

J. D. Williams 


Members 

L. J. Briggs 

R. COURANT 
J. H. Dellinger 
G. C. Evans 
L. M. Graves 


R. F. Mehl 
H. M. Morse 
P. M. Morse 


0. Veblen 


H. P. Robertson 
A. H. Taub 


CONTKACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS FOR THE 

APPLIED MATHEMATICS PANEL 


Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-444 

The Franklin Institute 
Philadelphia, Pa. 

Technical Representative, 

H. B. Allen 

Computations. 

OEMsr-618 

Columbia University 

New York, N. Y. 

Official Investigator, 

H. Hotelling 

Director : 

Allen Wallis 

Statistical methods applied to air combat analysis, torpedo 
tactics, acceptance inspection, research and development, 
and related problems. 

OEMsr-817 

University of California 
Berkeley, California 

Technical Representative, 

J. Neyman 

Statistical analysis applied to bombing research concerned 
with problems of land mine clearance, the theory of pattern 
bombing and the bombing of maneuvering ships, and the 
theory of bomb damage. 

OEMsr-818 

Columbia University 

New York, N. Y. 

Technical Representative, 

J. Schilt 

Mathematical and statistical studies of bombing problems; 
the application of IBM computing techniques to statistical 
problems in warfare analysis. 

OEMsr-860 

Princeton University 

Princeton, N. J. 

Technical Representative, 

S. S. Wilks 

Statistical methods applied to miscellaneous problems in war- 
fare analysis and to (1) verification of various long-range 
weather forecasting systems; (2) a study of fire effect tables 
and diagrams for warships; (3) bombing accuracy studies, 
analysis of guided missiles, and the performance of certain 
heat-homing devices; and (4) the clearance of mine fields 
by explosive devices. 

OEMsr-044 

New York L^niversity 

New York, N. Y. 

Technical Representative, 

R. Courant 

Investigations in shock wave theory. 

OEMsr-945 

New York University 

New A-ork, N. Y. 

Technical Representative, 

R. Courant 

Research in problems of the dynamics of compressible gases, 
hydrodynamics, thermodynamics, acoustics, and related 
problems. 

OEMsr-1007 

Columbia University 

New A'ork, N. Y. 

Technical Representatives, 

E. J. Moulton 

S. MacLane 

A. Sard 

Miscellaneous studies in mathematics applied to warfare 
analysis with emphasis upon aerial gunnery, studies of fire 
control equipment, and rocketry and toss bombing. 

OEMsr-1066 

Brown University 

Providence, R. 1. 

Technical Representative, 

R. G. D. Richardson 

Problems in classical dynamics and the mechanics of de- 
formable media. 

OEMsr-llll 

Institute for Advanced Study 
Princeton, N. J. 

Technical Representative, 

John von Neumann 

Studies of the potentialities of general-purpose computing 
equipment, and research in shock wave theory, with em- 
phasis upon the use of machine computation. 

OEMsr-1365 

Princeton University 

Princeton, N. J. 

Technical Representative, 

Merrill M. Flood 

Coordination of activities under Project AC-92 at the Uni- 
versity of New Mexico, Carnegie Institution of Washington 
at Pasadena, Columbia University, and Brown University. 



97 


CONTRACT NUMBERS, CONTRACTORS, AND SUBJECTS OF CONTRACTS FOR TllK 

APPLIED MATHEMATICS PANEL (ConUmicd) 

Contract 

Number 

Name and Address 
of Contractor 

Subject 

OEMsr-1379 

Northwestern University 

Evanston, III. 

Technical Representatives, 

E. J. Moulton 

Walter Leighton 

Studies in aerial gunnery, particularly the camera assessment 
of the performance of sights and of airplanes. 

OEIMsr-1381 

Carnegie Institution of Washington 
Pasadena, Calif. 

Technical Rei)resentative, 

Walter S. Adams 

Studies and experimental investigations in connection with 
the defensive fire power of various bomber formations by 
means of model planes with their guns replaced by suitable 
light sources, the total fire power being estimates of the 
light intensity. 

OEMsr-1384 

Harvard University 

Cambridge, Mass. 

Technical Representative, 

Garrett Birkhoff 

Studies of the principles which determine the dynamic be- 
havior of a projectile entering water and the application of 
these principles quantitatively to the prediction of under- 
water trajectories and ricochet. 

OExMsr-1390 

The University of New Mexico 
Albuquerque, N. M. 

Technical Representative, 

E. J. Workman 

Studies and e.xperimental investigations in collaboration with 
the Army Air Forces of the most effective formations and 
flight procedures for the B-29 airplane. Emphasis, originally 
upon the tactical use of the B-29, was later changed to a 
study of the defense of the B-29. 

Transfer of 

F unds 

National Bureau of Standards 

Computations by the Mathematical Tables Project for various 
agencies concerned with war research. 


98 


SEliVlCE PR()J]^CT NUMBERS 

The i)rojet‘ts listed below were transmitted to the Office of the Executive Secretary, ()SRD, from the War or Navy 
Department through either the War Dei)artment Liaison Officer for NDRC or the Office of Research and Inventions 
(formerly the Coordinator of Research and Development), Navy Dej^artment. 


Service 

Project Number 

Subject 

AC-27 

AC-91 

AC-92 

ARMY PROJECTS 

Design data for bombardier’s calculator. 

Statistical problems of combat bombing accuracy. 

Collaboration of the NDRC with the AAF in determining the most effective tactical application of the 
B-29 airplane (continuing under AAF Proving Ground Command, Fire Power Analysis Project). 

AC-95 

AC-109 

AC-115 

AC-122 

AN-23 

CE-33 

OD-143 

OD-179 

OD-181 

QMC-35 

QMC-38 

QMC-43 

SC-81 

SC-100 

SOS-2 

Analysis of Waller trainer film. 

Textbook on flexible gunnery. 

Study of data accumulated in sight evaluation tests. 

Study of gun camera film scoring in order to devise a scoring computer. 

Studies of HE-IB attack on precision target. 

Checking of hydraulic tables. 

Study of fuze dead-time correction in AA director. 

Statistical assistance in rocket propellant tests and specifications. 

Study of relative destructive effect of machine gun fire against airplane structures. 

Food storage data statistics. 

Studies of various statistical problems encountered at the Climatic Research Laboratory. 

Statistical consultation for Quartermaster Corps inspection service. 

Rapid solution of linear equations with up to twenty-six unknowns. 

Binomial distribution calculations. 

Probability theory of balloon barrages. 

N-110 

N-112 

NAVY PROJECTS 

Mathematical studies of lead-com})uting sights for use with gunnery training. 

Study and evaluation of sighting methods of instruction used in U. S. Naval Aviation free gunnery 
training. 

N-120 

NA-167 

NA-177 

NA-195 

Preparation of instruction course for quality control and statistically based sampling procedures. 

Study of nozzle design for jet motors. 

An analytical method of determining ships’ speeds in turns from photographs of ships’ wakes. 

Study of jet propulsion devices operating at subsonic and supersonic velocities (continuing under 
Contract NOa(s)-7370). 

ND-2 

NO-108 

NO-130 

NO-131 

Assistance to the Air Technical Division — studies of aircraft weapon effectiveness. 

Probability and statistical study of plane-to-plane fire. 

Air testing of Mark 15 bombsight. 

Probability studies desired in connection with estimating hits made by close-range AA gun fire at 
head-on airplane targets. 

NO-136 

NO-145 

NO-145 Ext. 

NO-145 Ext. 

NO-158 

NO-161 

Mathematical studies of dive-bomber and bomb trajectories in connection with Alkan dive-bombsight. 
Mathematical studies of bombing. 

Train probability calculations for bombing, November 1944. 

Probability curves for use in connection with gunnery salvo fire, June 1945. 

Antitorpedo-harbor defense nets. 

Theoretical studies of water entry phenomena (continuing under Contract NOa(s)-7370 with New 
York University and under Navy Contract with Harvard University). 

NO-188 

■ NO-206 

NO-237 

Study of torpedo spreads and their use against maneuvering targets. 

Studies of acceptance tests on ordnance material. 

Determination of depth of underwater explosions from surface observations (continuing under Con- 
tract NOa(s)-7370). 


99 


SERVICE PROJECT NUMBERS (Conlinucd) 


Service 

Project Number 

Subject 

NO-261 

Statistical analysis of the data on thermal characteristics of targets and the relative performance of 
candidate heat-homing equipment. 

NO-264 

NO-269 

NO-270 

Gun equilibrators. 

B-scan radar plotting device. 

Computation services (continuing under a transfer of funds from the Office of Research and Inven- 
tions to the Bureau of Standards). 

NO-272 

NO-280 

NO-294 

NR-101 

NR-105 

NS-165 

NS-166 

NS-302 

NS-364 

Computation of dynamic performance of AA computer (continuing under Contract NOrd-9153). 

Statistical assistance in rocket propellant tests and specifications. 

Study of tactical utilization of offset guns in fighter aircraft. 

Probability study of a proposed type of antiaircraft projectile. 

Fire effect tables (continuing under Contract NOrd-9240). 

Nonlinear mechanics. 

Gas globe phenomena in underwater explosions. 

High-temperature metals. 

Investigation of wave patterns created by surface vessels (continuing under Contract NOa(s)-7370). 


INDEX 


The subject indexes of all STR volumes are combined in a master index printed in a separate volume. For access to 
the index volume consult the Army or Navy Agency listed on the reverse of the half-title page. 


Aerial bombing, minefield clearance, 
84-87 

conclusions, 86 
early experiments, 86-87 
statistical method of analysis, 86 
Aiming error distribution, 11-12, 23 
Aiming error estimates, train-bombing 
combat data estimates, 39 
effect of misestimates on efficiency, 37 
misestimates, 37-39 
Aircraft allocation, large-scale bomb- 
ing, 29-30 

Aircraft torpedo attacks on warships, 
76-78 

Air-to-air bombing 

high attack from rear, 22 
low level frontal attack, 22 
scatter-bombing, 44-45 
AMP, NDRC, summary of work, 
1-6 

Angle of attack, train-bombing, 24 
Antitank mine clearance by linear ex- 
plosive devices, 79-82 
Applied Mathematics Panel, NDRC, 
summary of work, 1-6 
APQ-5 radar bombsight used in ship- 
bombing, 40, 41 
AZON (guided missile) 
pattern-bombing, 53-54 
probability of hits, 45 
probability study, 21-22 
scatter-bombing, 44-45 

Ballistic Res. Lab., Aberdeen Proving 
Ground, train-bombing prob- 
ability, 24 

Bangalore torpedo (mine clearance de- 
vice), 79 

Beach minefield clearance by rocket 
barrages, 83-84 

Blast bombs, mean area of effective- 
ness, 64-65 

Bomb fall, definition, 9 
Bomb patterns, uniform, 50-51 
Bombing, statistical studies, 58-6?, 
64-65 

see also Probability of hitting targets 
a ^priori and a posteriori investiga- 
tions, 12 

aiming error distribution, 11-12, 23 
choice of criterion, 9-10 
dispersion error distribution, 23 
mean area of effectiveness, large blast 
bombs, 64-65 

models used for calculations, 10-11, 
58 

probability of hitting target sections, 
58-62 

terminology and notation, 12-13 
Bombing methods 

aerial bombing for minefield clear- 
ance, 84-87 
air-to-air bombing, 22 


distinction between high- and low- 
altitude bombing, 39 
heat bombing, 88-92 
incendiary bomb attacks, 62-64 
pattern bombing, 46-57 
scatter bombing, 41-45 
single-release bombing, 14-22 
train-bombing, 23-45 
Bombing probability slide rules 
area-bombing, 60 

bomb-spacing, circular slide rule, 
31-34 

multiple-attack bombing, 35-36 
small- target bombing, 18, 60 
Bomb-spacing rules, train-bombing, 
31-35 

aiming-error statistic, 34 
bomb-dispersion statistic, 34 
calculator for optimum spacing, 31-34 
Bridge-bombing, 39-41 
British combat bombing operations, 23 

Carpet roll (minefield clearance device), 
79 

Cell diagram, graphical estimation of 
probability in bombing, 16-17 
Circular probable error (CEP) in bomb- 
ing statistics, 13, 18 
Circular targets, probability of hitting, 
14 

Crater effect, mine explosions, 82 

Demolition snakes (antitank minefield 
clearance device), 79-82 
Detonating cable (antipersonnel mine- 
field clearance device), 79 
Dispersion-error distribution, defini- 
tion, 23 

Elliptical targets, probability of hitting, 
14 

Expected number of hits, bombing 
studies criterion, 10 
Explosive pressure gauge, universal in- 
dicator mine, 79-82 

Felix (VB-6) heat-homing bomb, 91 
Fire control errors, standard deviation, 
72 

“Fire division” (vulnerability of build- 
ings to fire), 62-64 

Gauge for explosive pressure, universal 
indicator mine, 79-82 
Gaussian distribution, bombing statis- 
tics, 11-12 

Guided missile AZON 

see AZON (guided missile) 

Guided missile Razon, aiming point 
selection, 20 

Heat-homing devices, statistical analy- 
sis, 88-92 

(V 


conclusions, 90 
experimental set-up, 88 
Farrand instrument for thermal 
measurement, 88 
Felix bomb eye, 91 
improved type B device, 90-92 
problem, 88 

reliability of target signals, 90 
research recommendations, 92 
Hit probability 

see Probability equations ; Probability 
of hitting targets 

IBM equipment used in train-bombing 
probability calculations, 24 
Incendiary bomb attacks, 62-64 
“fire division” categories, 63 
M-47 bomb performance, 64 
M-50 bomb performance, 63, 64 
vulnerability of buildings to fire, 
62-64 

Lead angles for aircraft torpedo attacks 
on maneuvering warships, 76-78 
essential problem, 76 
formulae, 77-78 
numerical conditions, 76 
Linear explosive devices for minefield 
clearance, 79-83 
Bangalore torpedo, 79 
crater effect, 82 

curves of expected percentage of 
detonation, 82-83 
problem, 79 
skip effect, 82 

universal indicator mine, 79-82 

M-1 Snake (antipersonnel minefield 
clearance device), 79 
M-47 incendiary bomb, 64 
M-50, incendiary bomb, 64 
MAE (mean area of effectiveness), large 
blast bombs, 64-65 
Mark XXVII torpedo director, 69 
Mine clearance, statistical studies, 79-87 
aerial bombing, 84-87 
general discussion, 39-41 
linear explosive devices, 79-83 
pattern bombing, 54 
rocket barrages, 79, 83-84 
MPI (mean point of impact), bombing, 
11-12 

MRE (mean radial aiming error) bomb- 
ing, 13 

Multiple attack train-bombing, prob- 
ability, 28-31 

Offset, effect of mis-aiming, 36, 37 

Path clearance through mine fields, 
84-87 

Pattern-bombing, 46-57 

average proportion of hits, 46-47 

101 


102 


INDEX 


controlled-missile patterns, 53 
probabilities determined from opera- 
tional reports, 51-53 
probability of hits, 47-49 
probability of target coverage, 48-49 
regression equations, 53 
scatter-bombing, 41-45 
uniform bomb patterns, 50-51 
Pattern-bombing, synthetic patterns, 
54-57 

maneuvering targets, 54-55 
mine field clearance, 54 
photoelectric analyzer, 56-57 
Probability equations 

average proportion of hits, pattern 
bombing, 46-47 

circular target coverage, scatter 
bombing, 42 

density, aiming error distribution, 
11-12 

density, radial error distribution, 20 
expected hits on target, 10 
mean effective area, given bomb 
type, 65 

probability of at least n hits, 10 
probability of at least n hits, multiple 
attack train-bombing, 28, 49-50 
probability of hitting target with 
single bomb, 14 

proportion of target coverage, uni- 
form patterns, 49 
ship sinking, 41 

standard deviation, fire control errors, 
72 

statistically uniform distributed tar- 
get area, probability of hits, 
58-61 

torpedo salvo hits, 70-71 
Probability of hitting target with single 
bomb, 14-18 

approximations of probability, 15-16 
circular probable error, 18 
graphical estimation cell diagram, 
16-17 

slide rule for computing probability, 
18 

target areas, 14-15 
Probability of hitting targets, multiple 
attack train-bombing, 28-31 
aircraft allocation, 29-30 
calculations, 28 
conditions calculated, 28 
individual bomber effectiveness, 31 
optimum hits per aircraft lifetime, 32 
optimum number of aircraft per at- 
tack, 30-31 
slide rule, 35-36 

Probability of hitting targets, single 
attack train-bombing, 24-27 
angle of attack, 24 
maximum probability, 26 
optimum spacing, 24-25, 27 


RAZON (guided missile), aiming point 
selection, 20 

Rectangular targets, probability of 
hitting, 14 

Regression equations, pattern bombing 
statistics, 52-53 

Research recommendations for heat- 
homing devices, 92 
Right-triangle targets, 14 
Rocket barrages, beach mine field clear- 
ance, 79, 83-84 

factors determining effectiveness, 84 
probability of clearance with given 
number of barrages, 84 
rocket launching device, 83-84 

Saturation bombing, 58 
Saw-tooth zigzag torpedo path, 74-78 
Scatter-bombing, 41-45 
applications, 44-45 
theory, 41 

Serpentine torpedo path, 74-76 
Ship maneuverability in turns, 76 
Ship-bombing, 39-41 
hitting of ships, 41 
sinking of ships, 41 

Single attack train-bombing, proba- 
bility, 24-27 

Single-release bombing, 14-22 
aiming point selection, 18-21 
air-to-air bombing, 22 
application, 14 

imitation of combat errors, 20-21 
improvement of target coverage, 
18-20 

laterally controlled missile, AZON, 
21-22 

probability of hitting target, 14-18 
Skip effect, mine explosion, 82 
Slide rules for computing bombing 
probabilities 
area-bombing, 60 

bomb-spacing, circular slide rule, 
31-34 

multiple-attack bombing, 35-36 
small- target bombing, 18, 60 
Spacing bombs in train-bombing, 24- 
27, 31-35 

Spread angles for torpedo salvos 
see Torpedo salvos, spread angle 
Standard deviation 
aiming error, 11, 13 
bomb dispersion, 13 
fire control errors, 72 
Synthetic bomb patterns, 54-57 

Target areas, hit probabilities, 14 
Target in statistical studies, definition, 
9 

Taylor Model Basin, maneuverability 
characteristics of ships, 76 
Torpedo attacks on maneuverable tar- 
gets, 76-78 


Torpedo bombing, statistical studies 
lead angles for aircraft attacks 
on maneuvering warships, 76-78 
single-release bombing, 14 
terminology, 69 

torpedo salvos, spread angle, 69-73 
zigzag torpedo effectiveness, 73-76 
Torpedo salvos, spread angle, 69-73 
computational results, 72-73 
conclusion from studies, 73 
considerations affecting angle, 69-70 
probability of getting hits, 70-71 
problem, 69 i 

standard deviation of fire control ! 
errors, 72 

Train-bombing, 23-45 

aiming-error distribution, 23 
applications of theory, 39-41 
bomb spacing rules, 31-35 
dispersion-error distribution, 23 
efficiency compared with other bomb- 
ing methods, 36-37 
equipment problems, 23 
problem analysis, 23 
scatter-bombing, 41-45 
Train-bombing, applications, 39-41 
bridge bombing, 39-41 
mine fields, 39-41 
ships, hitting, 39-41 
ships, sinking, 39-41 
Train-bombing, probability, 24-31, 34- 
39 

aiming errors estimated from combat 
data, 37, 39 

comparison with other bombing 
methods, 36 
efficiency, 36-37 

mis-estimates of aiming-error, 37-39 
number of attacks necessary for 
maximum probability, 37 
offset, 36-37 

probabilities of hitting targets with 
multiple attacks, 28-31 
probabilities of hitting targets with 
single attacks, 24-27 
rules for determining probability, 
34-36 

Universal indicator mine (gauging de- 
vice for explosive pressure), 
79-82 

VB-6 (Felix) heat-homing bomb, 91 

Woofus, rocket launching device, 83-84 

Zigzag torpedo, effectiveness, 73-76 
numerical data, 73 
problem, 73 

saw-tooth torpedo path, 74-76 
serpentine torpedo path, 74-76 
unrestricted torpedo course, 73-76 




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